Background
In the famous Boltzmann's entropy formula, carved in the physicist's tombstone, a mysterious quantity $W$ appears (a):
$$S=k_B \log W\label{1}\tag{1}$$
We often hear that $W$ represents "the number of microstates corresponding to the macrostate of a system". However, this definition is problematic in classical physics, as noted for example by Fermi (bold is mine):
We have seen that a thermodynamical state of a system is not a sharply defined state of the system, because it cor- responds to a large number of dynamical states. This consideration led to the Boltzmann relation: $$S = k \log \pi$$ where $\pi$ is called the probability of the state. Strictly speaking, $\pi$ is not the probability of the state, but is actually the number of dynamical states that correspond to the given thermodynamical state. This seems at first sight to give rise to a serious difficulty, since a given thermodynamical state corresponds to an infinite number of dynamical states.
-E. Fermi, Thermodynamics (1937), Chapter VIII
Fermi then proceeds to propose a solution for this inconsistency in the framework of classical physics:
The phase space is divided into a number of very small cells all of which have the same hyper-volume $\tau$; the state is then characterized by specifying the cell to which the point representing the state belongs. Thus, states whose representative points all lie in the same cell are not considered as being different. This representation of the state of a system would evidently become exact if the cells were made infinitesimal. The cell representation of the dynamical states of a system introduces a discontinuity in the concept of the state of a system which enables us to calculate $\pi$ by the methods of combinatory analysis, and, hence, with the aid of the Boltzmann relation, to give a statistical definition of the entropy.
-E. Fermi, Thermodynamics (1937), Chapter VIII
This solution also has a problem, however, since the arbitrariness in the choice of $\tau$ leads to an arbitrary additive constant in $S$. Fermi claims that this inconsistency can ultimately be solved only by quantum mechanics:
[...] quantum theory introduces a discontinuity quite naturally into the definition of the dynamical state of a system (the discrete quantum states) without having to make use of the arbitrary division of the phase space into cells. It can be shown that this discontinuity is equivalent, for statistical purposes, to the division of the phase space into cells having a hyper-volume equal to $h^f$, where $h$ is Planck's constant ($h = 6.55 \cdot 10^{-27}$ cm$^2$ gm sec$^{-1}$) and $f$ is the number of degrees of freedom of the system.
-E. Fermi, Thermodynamics (1937), Chapter VIII
However, in modern statistical mechanics, it seems that the problem of the infinity of the microstates is not present (or ignored?)(b): in the microcanonical ensemble, entropy is simply defined as (c)
$$S= k_B \log \left(\int d^{3N}\mathbf p \ d^{3N} \mathbf q \ \rho({\{\mathbf p, \mathbf q\})} \right)\label{2}\tag{2}$$
where $\rho(\{\mathbf p, \mathbf q\})$ is the microcanonical probability density:
$$ \rho(\{\mathbf p, \mathbf q\})= \begin{cases} \text{const.} & E<\mathcal H (\{\mathbf p, \mathbf q\}) <E+\Delta E\\ 0 & \text{otherwise} \end{cases} \tag{3}\label{3} $$
In Eq.\ref{3}, $\mathcal H$ is the system's hamiltonian, $E$ is its energy and $\Delta E$ is chosen such that $\Delta E / E \ll 1$.
Eq.\ref{2} does not present the problem discussed by Fermi, since the integral appearing in it is just an hypervolume in phase space, which is finite if $E$ is finite. It appears therefore that Fermi's problem only appears if we (loosely?) define $W$ of Eq.\ref{1} as "the number of microstates of the system" without having a precise notion of what a microsatate is in statistical mechanics. However, this inconsistency is avoided if a definition like \ref{2} is chosen, where there is no mention of "microstates".
My question
What was the original interpretation Boltzmann gave of the $W$ appearing in Eq.\ref{1}?
Did he see it as a "number of microstates"? If so, was he conscious of the ambiguity of this concept, as pointed out by Fermi? Or did he see it, in more modern terms, as an hypervolume in phase space?
(a) Even if it appears that the equation was actually reformulated in this way later by Planck.
(b) But thr problem of the indeterminate additive constant remains.
(c) See for example K. Huang, Statistical Mechanics.