Poisson equation, stiffness matrix positive definiteness, Dirichlet boundary conditions

I have a question regarding the positive definiteness of the stiffness matrix. Specifically, I believe that it should be positive definite only when at least one Dirichlet point is given, so I would like a clarification on which is the precise moment that the stiffness matrix becomes positive definite. Let us consider a very simple model problem:

$$
-Delta u(pmb{x}) = f(pmb{x}), x in Omega
u(pmb{x}) = g(pmb{x}), pmb{x} in Gamma_D
partial_n u(pmb{x}) = 0, pmb{x} in Gamma_N.
$$

Using the divergence theorem this can be rewritten as:

$$
int_{Omega} nabla u cdot nabla v = int_{Omega}fv
u(pmb{x}) = g(pmb{x}), , pmb{x} in Gamma_D.
$$

A space $V_h subset mathcal{H}^1(Omega)$ is given, with basis functions $phi_1, ldots, phi_d, phi_{d+1},ldots,phi_{d+m}$. Let $Dir = {1,ldots,d}$ and $Ind = {d+1,ldots,d+m}$. The solution would then be of the form:

$$u_h(pmb{x}) = sum_{k=1}^{d+m}u_kphi_{k}(pmb{x}), pmb{x} in Omega,$$

and the linear system of equations that must be solved is:

$$
sum_{k=1}^{d+m}u_kint_{Omega}nablaphi_i cdot nablaphi_j = int_{Omega}fphi_i, , i in Ind
u_i = g_i, , i in Dir.
$$

The full stiffness matrix and right-hand side are:

$$
W_{ij} = int_{Omega}nablaphi_i cdot nablaphi_j, , i,j in Dircup Ind
f_i = int_{Omega}fphi, , i in Dir cup Ind.
$$

Then the system can be written more concisely as:

$$pmb{W}|_{Ind times Dircup Ind}pmb{u} = pmb{f}|_{Ind},$$

or if the Dirichlet columns are removed:

$$
pmb{W}|_{Ind times Ind}pmb{u}|_{Ind} = pmb{f}|_{Ind}-pmb{W}|_{Ind times Dir}pmb{g}.
$$

I know, that if even a single Dirichlet node is provided, then the system has a unique solution, which to me implies that $pmb{W}|_{Ind times Ind}$ is positive definite. However, if no Dirichlet node is provided, one is left with pure Neumann boundary conditions, and an additional constraint must be provided, for example (https://fenicsproject.org/olddocs/dolfin/2016.1.0/python/demo/documented/neumann-poisson/python/documentation.html):

$$int_{Omega}u = 0 implies sum_{k=1}^{m}u_kint_{Omega}phi_k = 0$$.

To me, requiring such an additional constraint implies that the matrix $pmb{W}|_{Ind times Ind}$ is not positive definite and that it has a zero eigenvalue. Yet, the answer here: In FEM, why is the stiffness matrix positive definite? claims that it is positive definite without referring to the number of Dirichlet nodes. Clearly I am missing something, and I would like to understand what.

Edit:

At first glance $pmb{W}|_{Ind times Ind}$ seems oblivious to Dirichlet nodes, since it includes only terms involving basis functions that correspond to non-Dirichlet nodes. However, introducing Dirichlet nodes modifies the non-Dirichlet basis functions (even non-Dirichlet basis functions have to vanish at a Dirichlet node, thus implicitly affecting the terms in $pmb{W}|_{Ind times Ind}$). Is anyone aware of a reference that makes this argument rigorous? More precisely a proof that the matrix $pmb{W}|_{Ind times Ind}$ becomes positive-definite upon the introduction of such a Dirichlet point?