First of all, your definition 1 is a bit faulty, you should say that $rank(df)_x$ is constant equal to $m=dim(M)$ at every point $x\in M$. With this in mind, these are equivalent definitions. However, I do not like either one of these definitions and for several reasons. Spivak's definition - because it depends on a nontrivial theorem (the immersion theorem), while a definition this basic should not depend on anything nontrivial. Also, for the reason that you stated. More importantly, I do not like both definitions - because they utterly fail in other, closely related situations. For instance, if I were to define the notion of a topological submanifold in a topological manifold along these lines, Spivak's will fail immediately (what is the rank of the derivative if I do not have any derivatives to work with?); Carroll's definition will fail because it will yield in some cases rather unsavory objects, like Alexander's horned sphere in the 3-space. The same if I were to use triangulated manifolds and triangulated submanifolds, algebraic (sub)varieties and analytic (sub)varieties.
Here is the definition that I prefer. First of all, what are we looking for in an $n$-dimensional manifold $N$ (smooth or not): We want something which is locally isomorphic (in whatever sense of the word isomorphism) to an $n$-dimensional real vector space (no need for particular coordinates, but if you like, just $R^n$). Then an $m$-dimensional submanifold should be a subset which locally looks like an $m$-dimensional vector subspace in an $n$-dimensional vector space. This is our intuition of a submanifold in any category (smooth, topological, piecewise-linear, holomorphic, symplectic, etc) we work with. Once you accept this premise, the actual definition is almost immediate:
Definition. Let $N$ be a smooth $n$-dimensional manifold. A subset $M\subset N$ is called a smooth $m$-dimensional submanifold if for every $x\in M$ there exists an (open) neighborhood $U$ of $x$ in $N$ and a diffeomorphism $\phi: U\to V\subset R^n$ ($V$ is open) such that $\phi(M\cap U)= L\cap V$, where $L$ is an $m$-dimensional linear subspace in $R^n$. (If you like coordinates, assume that $L$ is given by the system of equations $y_1=...y_{n-m}=0$.)
This is completely intrinsic. Next, you prove a lemma which says that such $M$ has a natural structure of an $m$-dimensional smooth manifold with topology equal to the subspace topology and local coordinates near points $x\in M$ given by the restrictions $\phi|(U\cap M)$. Then you prove that with this structure, $M$ satisfies the other two definitions that you know.
Remark. Note that this definition will work almost verbatim if I were to deal with topological manifolds: I would just replace "a diffeomorphism" with "a homeomorphism. If I were to work with, say, complex (i.e. holomorphic) manifolds, I would replace $R^n$ with $C^n$ (of course), use complex vector subspaces and replace "diffeomorphism" with
"a biholomorphic map". An so on.
Now, to the question why is it so much more complicated than the concept of a subgroup or a submodule or any other algebraic concept you can think of. This is because manifolds have a much richer structure. To begin with, they are topological spaces. (Notice that every submanifold is equipped with the subspace topology, so this has to be built in.) Then, the notion of vector spaces has to be used at some point. Next, there is the "local" thing (local charts)....
Best Answer
$\newcommand{\P}{\mathbf{P}}\newcommand{\R}{\mathbf{R}}$Caveat: The notion of "blow-up" described below is not the one you ask about, but it's doubtless similar in spirit. Perhaps this account will suggest how to define your notion precisely, or spur someone else to do so.
The "customary" algebro-geometric notion of blowing up amounts to the following: Let $i:M \hookrightarrow N$ be a smooth submanifold, and let $p:E \to M$ denote the normal bundle of $M$ in $N$. (You can think of the "normal bundle" merely as a complementary subbundle of $TM$ in $i^{*}TN$ if metrics are unavailable to take orthogonal complements.)
First, here's a description of blowing up one fibre of $E$ (i.e., "blowing up a point"). The set of lines through the zero vector in a fibre $E_{x}$ is a projective space $\P(E_{x})$. In the Cartesian product $\P(E_{x}) \times E_{x}$, consider the "tautological subset" $\widetilde{E_{x}}$ consisting of pairs $(\ell, v)$ such that $v \in \ell$. Projection on the second factor, i.e. $\Pi_{2}:\widetilde{E_{x}} \subset \P(E_{x}) \times E_{x} \to E_{x}$, induces a diffeomorphism except over the zero vector of $E_{x}$; the preimage of the zero vector is $\P(E_{x})$.
Geometrically, $\widetilde{E_{x}}$ comprises all one-dimensional linear subspaces of $E_{x}$, but now distinct subspaces have distinct zero vectors. In this sense, $\widetilde{E_{x}}$ is obtained from $E_{x}$ by removing the zero vector and gluing in the projective space $\P(E_{x})$; one point must be added for each line through the origin.
Here are pictures when $\dim(E_{x}) = 2$, created for Dana Mackenzie's What's Happening in the Mathematical Sciences, 2009; the labels indicate a complex vector space, but of course a real vector space is shown. Conceptually the real and complex pictures are identical.
To blow up the submanifold $M \subset N$, one shows the preceding construction can be made locally in $M$, i.e., over a coordinate neighborhood $U \subset M$, essentially by taking the Cartesian product of the preceding picture with $U$. Geometrically, $M$ is removed from $N$, and the projective bundle $\P(E)$ is glued in, in such a way that distinct normal directions at a single point of $M$ "touch different points of $M$" in the blow-up.
For example, the blow-up of $\R^{n} \subset \R^{n+k+1}$ may be viewed as $\R^{n} \times \widetilde{E}^{k}$, with $\widetilde{E}^{k} \to \R\P^{k}$ the tautological bundle of lines over the projective space.