Does the convergence of van Hove sequence equivalent to thermodynamic limit?

Missing a definition of the symbols you are reporting from the book cited in Definition 2, it is difficult to make more than some guess about the two definitions' mutual relations. It is highly probable that Definition 2 is about the thermodynamic limit in a system on a lattice.

However, there is a general observation, which is a partial answer to your question. The thermodynamic limit is a formal tool for recovering all the classical thermodynamic systems' properties from the ensemble formulation of statistical mechanics. The underlying physical idea is to look for the limit of some intensive version of thermodynamic potentials when the system's size grows. As such, its precise definition varies with the system and with the ensemble.

For example, the thermodynamic limit in the usual microcanonical, canonical, and grand canonical ensembles are:

• (microcanonical) $$ N \rightarrow \infty, V \rightarrow \infty, E \rightarrow \infty, $$ keeping constant $E/V$ and $N/V$.
• (canonical) $$ N \rightarrow \infty, V \rightarrow \infty $$ keeping constant $N/V$ at a fixed temperature.
• (grand canonical) $$ V \rightarrow \infty $$ at a fixed temperature and chemical potential.

Moreover, if additional or different extensive variables are required to describe a macrostate, they should be included as well.

The basic assumption which justifies the disappearance of finite size effects at the thermodynamic limit is that they would not scale like the volume $V$ but at least as $V^{\alpha}$, with $\alpha<1$. For a finite 3D system where particles interact via short-range forces, one would expect surface effects scaling like $V^{2/3}$. The difficult part is to show rigorously that such an expectation is actually satisfied by realistic interaction models.

I have nothing to add to @GiorgioP 's excellent answer, so I'll focus on issues specific to the book you address. (From your question, it is clear that several aspects I discuss below are clear for you; I nevertheless address them in case they might be useful to other readers.)

First, if $A\subset\mathbb{Z}^d$, then $|A|$ indeed denotes the number of vertices in $A$ (see the list of conventions, page xv of the free version of the book).

Spin systems vs fluids

The way you look at the thermodynamic limit applies better to a fluid, in which the number of molecules and the volume of the system are distinct quantities. The definition you state from our book applies to the Ising model, in which there is one spin per vertex, so the volume $V$ and the number of constituents $N$ would be equal in this case.

You'll find statements closer to those you are used to in the lattice-gas interpretation of this model. In this interpretation, a $+$ spin corresponds to a vertex occupied by a "particle", while a $-$ spin corresponds to an empty vertex. In this case, the number of $N$ of particles and the volume $V$ of the system are again distinct quantities and the thermodynamic limit takes the more usual form.

As @GiorgioP mentioned, you then have various notions of thermodynamic limits according to the ensemble you are considering. This is discussed in detail in Chapter 4 of the book. There is also an informal discussion in the introductory chapter (Section 1.3).

Convergence in the sense of van Hove

Finally, the precise condition you state (convergence in the sense of van Hove) is required if you want to prove convergence of the pressure (or of the thermodynamic potential relevant to the ensemble you consider) of finite-volume systems to a universal limit (that is, a limit that does not depend on the "shape" of the boxes in which your finite systems are contained or on the boundary condition used). You must then be able to show that boundary effects induce only higher-order corrections that disappear in the limit. For systems with short-range interactions, it turns out that these corrections scale like the size of the boundary of the system and will thus be negligible (remember that we are interested in densities) provided that the size of the boundary of the system grows slower than its volume, which is precisely what van Hove's condition guarantees.

Why is this usually not stated explicitly in Physics?

The answer is very simple: usually, in Physics, on works with nice sequences of "boxes" (cubes, for example), so this problem does not arise (or, better, the condition is automatically satisfied). This is also what is done in Chapter 4 of the book.