Question 1.
I have not seen it in the literature, but my knowledge is limited.
Question 2.
The idea to take $T_pM = (\mathbb R^m,\phi)$, where $\phi : U \to U'$ is a fixed chart around $p$, is nice. It means that you identify the tangent space $T_pM$ with $\mathbb R^m$ via a chart $\phi$ around $p$. This is completely legitimate, but I do not think that one should choose a specific $\phi$. Instead one should take into account all charts $\phi$ and "glue" the associated copies of $\mathbb R^m$ via the differentials of the transition functions. Let us make this precise. This will involve a lot of formalism and it is not as intuitive as you expect. The approach is purely requirements-based; its focus is to explicitly formulate our expectations regarding tangent spaces.
On an abstract level, we want to associate to each point $p$ of an $m$-dimensional manifold $M$ an $m$-dimensional tangent vector space $T_pM$ specified by the following requirements:
If $M$ is an open subset of $\mathbb R^m$, then $T_p M = \mathbb R^m$ (or more precisely, there exists a canonical isomorphism $T_pM \to \mathbb R^m$). This is a reasonable requirement: In any conceivable interpretation the set of all tangent vectors at $p$ should be nothing else than $\mathbb R^m$. It is the intuitive part of our approach.
The tangent space $T_pM$ is determined locally, i.e. $T_p M = T_pU$ for each open neigborhood $U$ of $p$. Again we can require more generally that there exists a canonical isomorphism $T_pU \to T_pM$.
Each chart $\phi : U \to U'$ around $p$ induces a specific isomorphism $d\phi_p : T_pM = T_pU \to T_{\phi(p)}U' = \mathbb R^m$.
Given two charts $\phi, \psi$ around $p$, we have $d\psi_p \circ d\phi_p^{-1} = d(\psi \circ \phi^{-1})_{\phi(p)}$. Here $\psi \circ \phi^{-1}$ is the transition map from $\phi$ to $\psi$ (which is a map between open subsets of $\mathbb R^m$) and $d(\psi \circ \phi^{-1})_{\phi(p)}$ is its standard derivative in the sense of multivariable calculus.
These requirements tell us how to construct $T_pM$. We denote by $\mathfrak C(M,p)$ the set of all charts $\phi : U \to U'$ on $M$ such that $p \in U$.
Let us begin by cumulating the vector spaces $T_{\phi(p)}U' =\mathbb R^m$ for all $\phi \in \mathfrak C(M,p)$. Formally this produces the set
$$\mathbb T_pM = \mathbb R^m \times \mathfrak C(M,p) .$$
The vector spaces $\mathbb R^m \times \{\phi\} \equiv \mathbb R^m = T_pU'_\phi$ and $\mathbb R^m \times \{\psi\} \equiv \mathbb R^m = T_pU'_\psi$ for $\phi,\psi \in \mathfrak C(M,p)$ will be identified non-trivially via the isomorphism $d(\psi \circ \phi^{-1})_{\phi(p)} : \mathbb R^m \to \mathbb R^m$. That is, on $\mathbb T_pM$ we define an equivalence relation by $(v,\phi) \sim (w,\psi)$ if $w = d(\psi \circ \phi^{-1})_{\phi(p)}(v)$. Now let
$$T_pM = \mathbb T_pM/\sim. $$
The equivalence classes with respect to $\sim$ are subsets of $\mathbb T_pM$ containing precisely one element of each $\mathbb R^m \times \{\phi\}$. Thus each inclusion $i_\phi : \mathbb R^m \to \mathbb T_pM, i_\phi(v) = (v,\phi)$, induces a bijection $\iota_\phi : \mathbb R^m = T_pU' \to T_pM, \iota_\phi(v) = [v,\phi]$. By construction
$$\iota_\psi \circ d(\psi \circ \phi^{-1})_{\phi(p)} = \iota_\phi \tag{$*$}$$
for all $\phi,\psi \in \mathfrak C(M,p)$.
Each bijection $\iota_\phi$ induces a unique vector space structure on $T_pM$ making $\iota_\phi$ an isomorphism and because or $(*)$ all these vector space structures on $T_pM$ agree.
Let us now check the above requirements.
Each open $M \subset \mathbb R^m$ has $id_M$ as a canonical chart, thus $\iota_{id_M} : \mathbb R^m \to T_pM$ is a canonical isomorphism which allows to identify $T_pM \equiv \mathbb R^m$.
Let $U \subset M$ be an open neighborhood of $p \in M$. Then $\mathfrak C(U,p) \subset \mathfrak C(M,p)$ and thus by definition we get a canonical isomorphism $j_U : T_pU = (\mathbb T_pU/\sim) \to (\mathbb T_pM/\sim) = T_pM$.
Given $\phi \in \mathfrak C(M,p)$, we take $d\phi_p = \iota_\phi^{-1} : T_pM \to \mathbb R^m$.
This is now obvious; it is a reformulation of $(*)$.
Let us remark that our definition of $T_pM$ is a special case of a direct limit construction.
What is the relation to the "derivatives of curves" approach in your question?
To each curve $u : J \to M$ through $p$ we can associate the equivalence class $$\theta(u) = [(\phi\circ u)'(0),\phi] \in T_p M $$
where $\phi \in \mathfrak C(M,p)$ is arbitrary. Its representative in the $\mathbb R^m \times \{\psi\}$-subset of $\mathbb T_pM$ is $((\psi\circ u)'(0),\psi)$. Thus the vector $u'(0) = \theta(u)$ subsumes all "localized derivatives" of $u$.
Clearly $\theta(u) = \theta(v)$ iff $u$ and $v$ are equivalent in the sense of your question. Thus we get a bijection between the set of equivalence classes of curves through $p$ and our $T_pM$. In other words, our $T_p M$ is indeed a variant of the tangent space defined by equivalence classes of curves through $p$, but the vector $u'(0) = \theta(u)$ is certainly closer to intuition than $[u]$.
The above definition of $T_pM$ also nicely shows how to endow the tangent bundle
$$T M = \bigcup_{p \in M} \{p\} \times T_p M$$
with a topology and a smooth structure. In fact, for each $\phi \in \mathfrak C(M,p)$ we get a canonical fiber-preserving bijection
$$\tau_\phi : \bigcup_{p \in U} \{p\} \times T_p M \to U \times \mathbb R^m, \tau_\phi(p,v) = (p,d\phi_p(v)) .$$
If $f : M \to N$ is smooth, we have the localized derivatives $d_{(\phi,\psi)} f_p = d(\psi \circ f \circ \phi^{-1})_{\phi(p)}$. Define
$$df_p = d\psi_{f(p)}^{-1} \circ d_{(\phi,\psi)}f_p \circ d\phi_p : T_pM \to T_{f(p)}N .$$
It is easy to verify that $df_p$ does not depend on the charts $\phi, \psi$. By definition the diagrams
$\require{AMScd}$
\begin{CD}
T_pM @>{df_p}>> T_{f(p)}N \\
@V{d\phi_p}VV @V{d\psi_{f(p)}}VV \\
\mathbb R^m @>{d_{(\phi,\psi)} f_p}>> \mathbb R^m \end{CD}
commute which nicely shows that the localized derivatives of $f$ at $p$ are not that bad as they appear at first glance. In fact, $df_p$ subsumes all these maps.
For a curve $u : J \to M$ through $p$ we get the derivative
$$du_0 : \mathbb R = T_0J \to T_pM .$$
By construction we have $du_0(1) = u'(0)$.
What about the isomorphism $d\phi_p : T_pM \to \mathbb R^m = T_{\phi(p)} U'$ which was defined above in a very formal way to identify $T_pM$ with $\mathbb R^m$? We have a commutative diagram
$\require{AMScd}$
\begin{CD}
T_pM =T_pU @>{d\phi_p}>> T_{\phi(p)}U' \\
@V{d\phi_p}VV @V{d(id_{U'})_{\phi(p)}}VV \\
\mathbb R^m @>{d_{(\phi,id_{U'})} \phi_p}>> \mathbb R^m \end{CD}
where the horizontal $d\phi_p$ is the derivative of $\phi$ at $p$ and the vertical $d\phi_p$ is our formal isomorphism. But
$$d_{(\phi,id_{U'})} \phi_p= d(id_{U'} \circ \phi \circ \phi^{-1})_{\phi(p)} = id$$
which shows that both interpretations of $d\phi_p$ are identical.
Finally, let us consider smooth maps $f : (M,p) \to (N,q)$ from $M$ to $N$ mapping a fixed $p \in M$ to a fixed $q \in N$. The function $\theta$ assigning to $f$ its derivative $df_p : T_pM \to T_qN$ is easily seen to be a surjection from the set of all smooth maps $(M,p) \to (N,q)$ to the vector space of all linear maps $T_pM \to T_qN$. Clearly $\theta(f) = \theta(g)$ defines an equivalence relation $f \sim g$ for smooth maps $(M,p) \to (N,q)$. As for tangent vectors one can therefore identify $df_p$ with the equivalence class $[f]$ of $f$ which results in an alternative definition of the derivative.
This is precisely what is done for the cotangent space $T^*_pM$: The derivative $df_p : T_pM \to T_{f(p)}\mathbb R = \mathbb R$ is interpreted as the equivalence class $[f]$.
Remark 1:
Moishe Kohan comments that the above construction of $T_pM$ is closely related to an alternative description of the tangent bundle $TM$. This can be obtained by gluing all trivial bundles $B_\phi = U \times \mathbb R^m$ over the domains of charts $\phi : U \to U'$ via the transition maps of the fibers. Technically this is done via the maps $\tau_{(\phi,\psi)}: U \cap V \to GL(\mathbb R^m),\tau(p) = d(\psi \circ \phi^{-1})_{\phi(p)}$. The fiber $T_pM$ of the "glued bundle" $TM$ is thus obtained by gluing the collection of all $\{p\} \times \mathbb R^m_\phi = \{p\} \times \mathbb R^m$ with $p$ in the domain of $\phi$ via the transition maps $d(\psi \circ \phi^{-1})_{\phi(p)}$ (note that $p$ is always in the common part $U \cap V$ of all charts $\phi, \psi$ around $p$). This is in fact nothing else than the above construction.
Remark 2:
One more alternative construction of $T_pM$ goes as follows. Consider the vector space $\mathbf T_pM = (\mathbb R^m)^{\mathfrak C(M,p)} = \prod_{\phi \in \mathfrak C(M,p)} \mathbb R^m$ and define
$$T_pM =\{(v_ϕ)∈\mathbf T_pM \mid d(\psi \circ \phi^{-1})_{\phi(p)}(v_ϕ)=v_ψ \text{ for all } ϕ,ψ \in \mathfrak C(M,p)\}.$$
It is easy to check that is a linear subspace of $\mathbf T_pM$. The product projections $\pi_\phi: \mathbf T_pM \to \mathbb R^m$ restrict to linear maps $d\phi_p : T_pM \to \mathbb R^m$ which are easily seen to be isomorphisms. Verification of 1. - 4. is straightforward.
Let us remark that this definition of $T_pM$ is a special case of an inverse limit construction.
Best Answer
Yes, it's correct, except for the technical point that the derivative map acts on (and maps to) the tangent space, not the affine tangent space. The tangent vector $u'(0)$ is based at the origin, not at $u(0)$, unless you are totally reinventing notation. But it's easy enough to work with the usual tangent space and just say that its affine translation is the best (affine) linear approximation at $p$.
You can prove the result by taking a local smooth extension of $f$ (in a neighborhood of $p$) to an open set in $\Bbb R^M$, as is quite standard. The main point then is that $$x-p = \pi(x-p) + (x-p)^\perp,$$ where $u^\perp\in (T_pM)^\perp$, and $\|(x-p)^\perp\| = o(\|x-p\|)$. You can verify this by assuming $T_pM = \Bbb R^m\times\{0\}\subset\Bbb R^M$ [as I said in the comments, horrendous, horrendous choice of notation] and writing $M$ locally as the graph of a smooth function $\phi\colon U\to\Bbb R^{M-m}$, $0\in U\subset\Bbb R^m$ open, $p=0\in\Bbb R^M$; the claim then follows by considering the second-order Taylor polynomial of $\phi$ at $0$.
The uniqueness result follows from the fact that the derivative $df_p$ is equal to the restriction of $dF_p$ for any local smooth extension $F$ of $f$.