Article: "the" or "a"

Hello, this is my sentence:

If I had known there were so many problems with this apartment, I would have never signed the/a lease with you.

This place is too noisy. The reason why I signed the/a lease with you in the first place is to move into a quiet place.

if I understood my teacher's explanation, I can use either article in the examples above without any considerable change to the meaning, but there is a slight nuance:
the lease: the one that I signed. The emphasis is on the lease that I signed.
a lease: what goes for the lease that I did sign would have applied to any other lease agreement with that property manager that I might have signed but did not. The emphasis is less on the lease than on the signing.

Have I learned my lesson well/did I understand the teacher correctly? (I would go with "the" for both, but either is correct).