First, you also have to check all possible separating equilibria. You only checked (MSc,BSc), not (BSc,MSc) with all possible reactions of the receiver.
No separating equilibrium exists here.
The pooling eq you find is correct. There is no pooling eq in which both types play MSc.
The intuitive criterion is a refinement to rule out "unreasonable" ...
If the borrower goes bankrupt, the lender has still received all of the loan payments up to that point, and the borrower's remaining assets are also distributed amongst its lenders (creditors). If there is only one lender they will get all the assets.