Landau free energy, ising mean field and the "full partition function". Discrepancy between two similar approaches From what I understand, for example, in the neighbor interactions Ising model, we can write the partition function as:
$$Z = \sum_{m}\Omega(m)e^{-\beta E(m)}=\sum_{m}e^{-\beta \tilde F(m)}=e^{-\beta F}\tag 1$$
where $\Omega(m)$ is the number of configurations corresponding to a magnetization $m$, $\tilde F(m)$ is a "partial" free energy corresponding to $m$. It's what is approximated using the landau free energy and $F$ is the free energy of the system.
Now, the mean field of the ising model is:
$$H_{MF}=\frac{NqJm^2}{2}-qJm\sum_{i=1}^Ns_i \tag 2$$
And summing over all configurations we get:
$$Z_{MF}=\prod_{i=1}^N\sum_{s_i=\pm 1}e^{-\beta H_{MF}}=e^{-\frac{\beta NqJm^2}{2}}[2\cosh(\beta qJm)]^N \tag 3$$
$F(m) = -T\log(Z_{MF}(m))$ definitevely gives a variational free energy (the equivalent of $\tilde F$ in equation (1) and not F). Using this, we can find the self-consistency equation, for example by minimizing $F(m)$ with respect to $m$.
I've also seen an other way to do this mean field, with this hamiltonian (for example in the statistical field theory lecture of David Tong: 1.1.2 Mean Field Theory ).
$$H'_{MF}=\frac{NqJm^2}{2}-qJNm^2= -NqJm^2/2\tag 4$$
where it looks like $m$ is taken to be equal to $\frac{1}{N}\sum_{i}s_i$ which seems a bit odd since I believe that $m$ is supposed to be a thermal average not an average over "sites". Anyway we can write the partition function as the sum over all the configurations (as before) which is now a sum over possible magnetizations:
$$Z'_{MF} = \sum_{m}\Omega(m)e^{\beta NqJm^2/2}\tag 5$$
Again, following David Tong, we can find $\Omega(m)$ and so we find that:
$$Z'_{MF}=\sum_{m}\exp(-\beta (-NqJm^2/2 - T(\log(2)-(m+1)\log(m+1)/2-(1-m)\log(1-m)/2)))=\sum_{m}e^{-\beta \tilde F(m)}\tag 6$$
In this case $-T\log(Z'_{MF})$ doesn't give a variational free energy (it's the "full free energy"), the variational free energy is (as expected from (1)) the big expression in the exponential. And with $\tilde F(m)$ we can find the same self-consistency equation as before.

I'm confused, these two approaches seem to me to be the same, and yet: $-T\log(Z_{MF})$ is a variational free energy and $-T\log(Z'_{MF})$ isn't (it's the "full free energy"). In the first case, the variational free energy is $F(m)=-T\log(Z_{MF})$ while in the second case, the variational free energy is  $\tilde F(m)\neq-T\log(Z'_{MF})$.
When I first saw the mean field approach, I convinced myself that the partition function given by $H_{MF}$ would then give a variational free energy. This is the case for $Z_{MF}$ but not for $Z'_{MF}$.
So, is a mean field approach supposed to give us a partition function dependent upon an order parameter, here $Z_{MF}(m)$? Or is it supposed to gives us a full partition function, here $Z'_{MF}$? Why these two similar/mean field approaches give differents objects at the end?
I've always wonder why we do we consider that $m$ was constant when we evaluate the partition function (3) even though we are summing over all value of $s_i$ (and $m$ and $s_i$ are linked, in all case, I suppose the self-consistency equation takes care of that part?). Is this the reason why the first approach yields a partition function dependent upon $m$, because we are working with only one value of $m$ when we do the sum (but it seems kind of weird, as long as we sum over all possible states, it should be equivalent to a sum over all possible magnetizations)?
 A: 
[...] it looks like m is taken to be equal to $\frac{1}{N}\sum_i s_i$ which seems a bit odd since I believe that m is supposed to be a thermal average not an average over "sites".

Yes, that's right.  The $m$ which appears at that section of Tong's notes is not the thermal average magnetization, but rather the average magnetization of a particular configuration of spins.
For clarity, let's use the symbol $m$ to mean the thermal average magnetization and $\hat m = \frac{1}{N}\sum_i s_i$ to mean the average magnetization of a particular configuration $\{s_i\}$.  From the definition of the free energy, we have that $Z = e^{-\beta F}$.  However, in the mean field approach we can write
$$Z = \sum_{\{s_i\}}e^{-\beta E(\{s_i\})}\rightarrow \sum_{\hat m}\Omega(\hat m) e^{-\beta E(\hat m)} = \sum_\hat m e^{-\beta \hat F(\hat m)} $$
where we define $\hat F(\hat m):= E(\hat m) - \frac{1}{\beta}\log\big(\Omega(\hat m)\big)$.
In the thermodynamic limit, $e^{-\beta \hat F(\hat m)}$ is an extremely sharply-peaked function about its maximum $e^{-\beta \hat F_0} \equiv e^{-\beta\hat F(\hat m_0)}$, where $\hat m_0$ minimizes $\hat F$.  As a result, we can say that
$$e^{-\beta F} = \sum_{\hat m} e^{-\beta \hat F(\hat m)} \approx e^{-\beta F_0}$$
and that $m \approx \hat m_0$.  It is in this sense that the two approaches you refer to produce the same result in the limit of large $N$.
