Zhen, I am not sure I understand your question.
Every stict 2-limit is obviously also a 1-limit in the underlying 1-category, so these are not really different concepts (a 2-limit is a strengthen version of a limit; BTW, since Cat is 2-complete then every 1-limit in Cat is automatically a 2-limit).
Your construnction of a 2-pseudo pullback is fine. However, it is easy to verify that it is not stable under equivalence of categories ([Added] in the sense that: $C$ is a limit of $F$ and $D$ is equivalent to $C$ does not imply that $D$ is a limit of $F$). All of the mentioned limits are defined in terms of "strict" adjunctions (or, more acurately, in terms of strict universal properties), i.e. there is a natural isomorphism:
$$\Delta(C) \rightarrow F \approx C \rightarrow \mathit{lim}(F)$$
To obtain a concept that is stable under equivalences, you have to replace this natural isomorphism by a natural equivalence of categories (plus perhaps some additional coherence equations). Of course, every strict 2-limit is also a "weak" 2-limit in the above sense (because every isomorphism is an equivalence), so again in a complete 2-category you will not get anything new.
[Added]
3. Let $\mathbb{W}$ be a 2-category, and $X$ a 1-category. There are three types of 2-cateogrical cones in $\mathbb{W}$ of the shape of $X$:
$\mathit{Cone}$ --- objects are strict functors $X \rightarrow \mathbb{W}$, 1-morphisms are strict natural transformations between functors, and 2-morphisms are modifications between natural transformations
$\mathit{PseudoCone}$ --- objects are pseudo functors $X \rightarrow \mathbb{W}$, 1-morphisms are pseudo natural transformations between functors, and 2-morphisms are modifications between natural transformations
$\mathit{LaxCone}$ --- objects are lax functors $X \rightarrow \mathbb{W}$, 1-morphisms are lax natural transformations between functors, and 2-morphisms are modifications between natural transformations
A limit of a strict functor $F \colon X \rightarrow \mathbb{W}$ is a 2-representation of:
$$\mathit{Cone}(\Delta(-), F)$$
where $\Delta$ is the usual diagonal functor. A pseudolimit of $F$ is a representation of:
$$\mathit{PseudoCone}(\Delta(-), F)$$
And a lax limit is a representation of:
$$\mathit{LaxCone}(\Delta(-), F)$$
In each case if you take equivalent functors, then you get equivalent representations. However, in each case the notion of equivalent functors is different. Perhaps your problem is that you are using the equivalence from $\mathit{PseudoCone}$ in the context of $\mathit{Cone}$
[Added^2]
I have missed one of your questions:
Finally, there is the non-strict 2-pullback, which as I understand it has the same universal property as the pseudo 2-pullback but with "unique functor" replaced by "functor unique up to isomorphism".
If by a non-strict pullback you mean a weak (pseudo)pullback in the above sense, then the universal property is much more subtle --- it does not suffice to say that there is a functor $f \colon X \rightarrow \mathit{Lim}(F)$ that is unique up to a 2-isomorphism (just like in the definition of a limit you do not say that there is an object which is unique up to 1-isomorphism), you have to say that for every cone $\alpha \colon \Delta(X) \rightarrow F$ there exists $f \colon X \rightarrow \mathit{Lim}(F)$ such that for any cone $\beta$ on $X$ with its $g \colon X \rightarrow \mathit{Lim}(F)$ and every family of 2-morphism $\tau \colon \alpha \rightarrow \beta$ that is compatible with $F$ there exists a unique 2-morphism $f \rightarrow g$ such that everything commutes.
However, if by a non-strict pullback you mean a lax pullback, then the construction is similar to your construction of a pseudopullback --- without requirement that your $f$ and $g$ are isomorphisms.
You have aslo asked:
Where can I find a good explanation of strict 2-limits / pseudo 2-limits / bilimits and their relationships, with explicit constructions for concrete 2-categories such as $\mathfrak{Cat}$? So far I have only found definitions without examples. (Is there a textbook yet...?)
I do not know of any good textbook, but can provide you with two examples.
There is a simple general procedure to construct strict/pseudo/lax limits and colimits in $\mathbf{Cat}$. You shall notice that to give a monad is to give a lax functor $T \colon 1 = 1^{op} \rightarrow \mathbf{Cat}$. Then the lax colimit of $T$ is the Kleisli category for the monad $T$, and the lax limit of $T$ is the Eilenberg-Moore category for the monad $T$. This idea may be pushed a bit further: you may think of a lax functor $\Phi \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$ as of a kind of a "multimonad". Then its multi-Kleisli resolution is given by the Grothendieck construction $\int \Phi$ (this construction gives a fibration precisely when $\Phi$ is a pseudofunctor). And similarly its multi-Eilenberg-Moore category is given by a suitable collection of (ordinary) algebras. In these construction if you impose a reguirement on cartesian morphisms / algebras to be isomorphisms, then you get a pseudocolimit / pseudolimit of $\Phi$, and if you impose identities instead of isomorphisms you get a colimit / limit.
What is more, the Grothendieck construction works also for the bicategory of distributors; and because there is a duality on the bicategory of distributors you may construct (lax/pseudo/strict) limits in this bicategory via the Grothendieck construction as well.
The doom and gloom here comes from using the wrong notion. Note that limits and colimits of topological spaces are not homotopically invariant (that is, they aren't well-defined operations on homotopy classes of diagrams of spaces), so for the purposes of homotopy theory they aren't always the right thing to do. You should instead be looking at homotopy limits and homotopy colimits, which are in a suitable sense the "derived functors" of limits and colimits; in particular, they are homotopically invariant constructions.
Here the news is much better. The nicest statement comes when we avoid basepoints and instead replace the fundamental group with the fundamental groupoid $\Pi_1(X)$, which is a (higher) functor from spaces to the $2$-category of groupoids, functors, and natural transformations. This is important because $2$-categories also have a notion of homotopy limits and colimits, also called $2$-limits and $2$-colimits. And now I claim that
$\Pi_1$ preserves all homotopy colimits.
This is an abstract and high-powered version of the Seifert-van Kampen theorem (although admittedly it's only useful to the extent that you can actually compute homotopy colimits). Unlike the usual Seifert-van Kampen theorem, it is powerful enough to allow you to compute the fundamental group of $S^1$ by decomposing it as two intervals which intersect in two points; the point is that working with fundamental groupoids gives you the freedom to use more than one basepoint. This argument should be somewhere in Brown's Topology and Groupoids.
Morally this result is true because $\Pi_1$ is the homotopy left adjoint to the "forgetful functor" sending a groupoid $\Pi$ to its classifying space $B\Pi$ (a mild generalization of the construction of Eilenberg-MacLane spaces). The ordinary Seifert-van Kampen theorem can be thought of as giving conditions under which it is possible to compute a homotopy pushout in spaces as an ordinary pushout.
Example. Let $G$ be a group acting on a path-connected space $X$ preserving a basepoint $x$. The ordinary quotient $X/G$, or $X_G$, is a prime example of a colimit which is not a homotopically invariant operation: in general the homotopy type of the quotient is very sensitive to exactly how $G$ acts on $X$, and we cannot replace the action by a homotopy equivalent action. In particular the fundamental group of $X/G$ is not determined from the data of the action of $G$ on $\pi_1(X, x)$.
The homotopically invariant replacement is the homotopy quotient or Borel construction, variously notated $X//G$ or $X_{hG}$, and given explicitly by
$$X \times_G EG$$
where $EG$ is a contractible space on which $G$ acts freely and $\times_G$ means to take the quotient of the product by the diagonal action of $G$. The idea is that $X \times EG$ is a "resolution" of $X$ as a $G$-space suitable for computing the "derived functor" of taking quotients.
The abstract Seifert-van Kampen theorem above now implies that the fundamental group of $X//G$ is the homotopy quotient of $\pi_1(X, x)$ by the action of $G$. Once you work out what homotopy quotients are for groupoids, this turns out to be precisely the semidirect product
$$\pi_1(X, x) \rtimes G.$$
Something more complicated happens if you don't assume that $G$ preserves a basepoint or if you use the more general and homotopically correct definition of "action of $G$," but that's another story.
Best Answer
A pullback as just a certain subset of the product: those pairs $(a,b)$ on which the two given maps on coordinates agree, i.e. $f(a)=g(b)$.
This needs an explicit example, which I'll build on the lovely product diagram we already have available. So map $X=\{1,2,3,4,5,6\}\to \{\text{even},\text{odd}\}$ in the obvious way, and $Y=\{\clubsuit,\diamondsuit,\heartsuit,\spadesuit\}$ by saying, for instance, the black suits $\clubsuit,\spadesuit$ are "even" and the others are "odd." Then the pullback becomes a subset of $X\times Y$ given by pairs $(n,s)$ where $n$ and $s$ have the same image in $\{\text{even},\text{odd}\}$. That is, the pullback has the twelve elements $\{(1,\heartsuit),(1,\diamondsuit),(3,\heartsuit),...,(5,\diamondsuit),(2,\spadesuit),(2,\clubsuit),...,(6,\clubsuit)\}$. If this begins to make some sense, see if you can compute