I will speak about $\mathbf{Cat}$ as an ordinary locally small category. There is no significant difference between the the various notions because filtered colimits in $\mathbf{Cat}$ preserve equivalences, and so a category is finitely presentable in the bicategorical sense if and only if it is equivalent to one that is finitely presentable in the enriched sense, and because the 2-categorical structure of $\mathbf{Cat}$ comes from cartesian closedness, this is also the same as finite-presentability in the ordinary sense.
There are two obvious necessary conditions for a category to be finitely presentable: it must have only finitely many objects and only countably many morphisms. Since the "delooping" functor from monoids (resp. groups) to categories preserves filtered colimits, a monoid (resp. group) that is finitely presentable as a category must be finitely presentable as a monoid (resp. group). Thus there are categories with finitely many objects and countably many morphisms that are not finitely presentable. Similarly, there are finitely generated categories that are not finitely presentable.
It is straightforward to show that a colimit for a finite diagram of finitely presentable objects is again a finitely presentable object. Consider the smallest full subcategory $\mathbf{Cat}_\mathrm{fp}$ of $\mathbf{Cat}$ that is closed under isomorphisms, finite colimits and contains $\mathbb{1}$, $\mathbb{2}$, and $\mathbb{3}$. By construction, every object in $\mathbf{Cat}_\mathrm{fp}$ is a finitely presentable category. On the other hand, since $\{ \mathbb{1}, \mathbb{2}, \mathbb{3} \}$ is a dense generating set for $\mathbf{Cat}$, every finitely presentable category must be in $\mathbf{Cat}_\mathrm{fp}$. Thus, we have an inductive characterisation of finitely presentable categories.
One can also characterise finitely presentable categories using generators and relations as in the classical cases. Let us say a finitely presented category is one that has a presentation in terms of finitely many objects, finitely many morphisms, and finitely many equations. The usual methods can be used to show that a finitely presented category is finitely presentable in the abstract sense. And it is clear that every category is a colimit for a directed diagram of finitely presented categories, so it follows that every finitely presentable category is a retract of a finitely presented category. But again the usual methods show that a retract of a finitely presented category must also be finitely presented, so the two notions coincide.
Going back to $\{ \mathbb{1}, \mathbb{2}, \mathbb{3} \}$ again, consider the induced Yoneda representation $N : \mathbf{Cat} \to [\mathbf{\Delta}_{\le 2}^\mathrm{op}, \mathbf{Set}]$, where $\mathbf{\Delta}_{\le 2}$ is the full subcategory of $\mathbf{Cat}$ spanned by $\{ \mathbb{1}, \mathbb{2}, \mathbb{3} \}$. The functor $N$ is fully faithful, preserves filtered colimits and all limits, so it has a left adjoint, say $L : [\mathbf{\Delta}_{\le 2}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Cat}$, and this must send finitely presentable objects in $[\mathbf{\Delta}_{\le 2}^\mathrm{op}, \mathbf{Set}]$ to finitely presentable objects in $\mathbf{Cat}$. But a finitely presented object in $[\mathbf{\Delta}_{\le 2}^\mathrm{op}, \mathbf{Set}]$ is precisely a presheaf that is componentwise finite, so we may deduce that a category that has only finitely many morphisms is finitely presentable. Of course, this could also be proved using the fact that finitely presented categories are finitely presentable, but it is nice illustration of the general fact that the left adjoint of an $\aleph_0$-accessible functor must preserve finite presentability.
I suppose I should give an interesting example of a finitely presented category. There is a finitely presented category $\mathbb{T}$ equipped with a finite collection of finite cones such that the category of groups is (canonically) equivalent to the category of all functors $\mathbb{T} \to \mathbf{Set}$ that send the given cones to product cones. More generally, the same is true for any algebraic theory with finitely many operations and finitely many axioms. One thinks of this $\mathbb{T}$ as a finitely presentable approximation to the full Lawvere theory.
I've modified your notation slightly. $x_0$ is the point instead of $x$, and $\psi$ is indexed by pairs $x,y$ in $D$.
To show universality, you want to suppose that for any set $S$ with a cocone of maps $\psi_{x,y}' : \newcommand\Hom{\operatorname{Hom}}\Hom(x_0,x)\to S$, you can construct a unique map $\alpha : F(x_0)\to S$.
Then given $y\in F(x_0)$, define $\alpha(y)=\psi_{x_0,y}'(\newcommand\id{\operatorname{id}}\id_{x_0})$.
Then we check that $\alpha \circ \psi_{x,y} = \psi'_{x,y}$.
Suppose $g\in \Hom(x_0,x)$, then $\psi_{x,y}(g) = F(g)(y)$.
Then $$\alpha(\psi_{x,y}(g)) = \alpha(F(g)(y))=\psi'_{x_0,F(g)(y)}(\id_{x_0}).$$
Observe that given $g\in \Hom(x_0,x)$, $g$ defines a morphism between the pairs $(x,y)$ and $(x_0,F(g)(y))$ by definition, so we must have $\psi'_{x_0,F(g)(y)} = \psi'_{x,y}\circ g_*$.
Thus $\alpha(\psi_{x,y}(g)) = \psi'_{x,y}(g_*\id_{x_0}) = \psi'_{x,y}(g)$, as desired.
As for uniqueness, suppose $\beta$ were another appropriate map $F(x_0)$ to $S$.
Then since $y = \id_{F(x_0)}y = F(\id_{x_0})y = \psi_{x_0,y}(\id_{x_0})$, for any $y\in F(x_0)$, we have $\beta(y) = \beta(\psi_{x_0,y}(\id_{x_0}))=\psi'_{x_0,y}(\id_{x_0}) = \alpha(y)$. Thus $\alpha$ is unique.
Best Answer
Let $f : Y_1 \to Y_2$ be a morphism in $\mathsf C$. Considering the object $\lim \mathsf C(F-,Y_1)$ as a constant functor, there is a natural transformation $$\lim \mathsf C(F-,Y_1) \longrightarrow \mathsf C(F-,Y_1) \stackrel{f_*}\longrightarrow \mathsf C(F-,Y_2)$$ (the first one is precisely the cone from $\lim \mathsf C(F-,Y_1)$ to $\mathsf C(F-,Y_1)$, and the second one is the natural transformation whose components are pre-composition by $f$) and so, the universal property of $\lim \mathsf C(F-,Y_2)$ tells us that there is a unique morphism $f_\flat : \lim \mathsf C(F-,Y_1) \to \lim \mathsf C(F-,Y_2)$.