I have been thinking about this issue a lot recently after reading Cochrane's excellent textbook. After some struggles, I think the key to understand all this is to pay attention to the subscripts, because they are important. Here is my take on your question.
I will use the form as so:
$$r^e_{i,t} = \alpha_i+\beta_{i,MKT}(r_{MKT,t}-r_{f,t})+\beta_{i,SMB} r_{SMB,t}+\beta_{i,HML}r_{HML,t}+\epsilon_{i,t}$$
This is population regression. You can think of projecting excess return on to the space of the factors and a constant. If you read the theory part seriously, you can see this is how we define the $\beta$s in the expected return-beta form of any linear factor model.To think about this relationship, you can fix i (or the vectorized form $r^e_t=[r^e_1,\cdots,r^e_N]'$), then we can think of the regression above as a time-series regression as
$$r^e_{t} = \alpha+\beta_{MKT}(r_{MKT,t}-r_{f,t})+\beta_{SMB} r_{SMB,t}+\beta_{HML}r_{HML,t}+\epsilon_{t}$$
This is the usual linear regression you see in any textbook. By construction, the $E[\epsilon_t|factors]=0$.
The more interesting thing is to think about what we are doing when we look at the intercept to see if the model does a good job pricing our left hand side portfolio.
First of all, when you talk about a factor model, the main implication of it is actually on the expected return not on return.
$$E[r^e_i] = \beta_i \lambda$$.
where $\lambda$ is the risk premium associated with your factor. There are two things we need to pay attention to.
- The subscript doesn't involve t, because we are dealing with an unconditional model which we think of the realization of returns and beta at each time as drawn from a distribution of $r^e_i$ and $\beta_i$ for each firm.
- You may wonder where is the factor in this model. The factor only enters into the model through its risk premium. It's only in the special case when your factors are excess returns, the risk premium $\lambda=E[f]$.
Now with these concepts clear up, we can proceed to understand Fama French 3-factor model.So what they propose is that the 3 factors are three long-short portfolio returns, namely MKT, SMB and HML. Given what I have talked about above, the model (APT, ICAPM) essentially says:
$$E[r^e_i] = \beta_{i,MKT}E[r_{MKT}-r_f]+\beta_{i,SMB} E[r_{SMB}]+\beta_{i,HML} E[r_{HML}]$$
Remember, these relationship are predicted by the model to hold in the population. What we observe is only the historical realization of these returns, so we need to estimate this relationship.
Finally, let's relate back to the time-series regression of Fama French at the beginning. By taking the unconditional expectation of both sides of the first equation and compared to the theoretical model above. You can see the implication for those MKT, SMB and HML being factors that perfectly price assets is that $\alpha_i =0$ $\forall i$ jointly. To test this, we can use the GRS(1989) test statistics.
The question I have, is there are difference between these or are all the same? Primarily I am interested in the explanation for the position of the Alpha and the risk free rate, I also notice that the error term is sometimes noted sometimes not. Is it a mere question of preference or are there deeper implications as to how you write it? I thank you for any feedback! 
