You can discover when you return to the vault that the Enclave requested access to it because the password doesn't match their records.

What reason could they have to try to open it?

Your Father.....

Nope.The entry on the overseer's terminal only appears after the Enclave takes over the purifier and he dies,it's not there at the time you escape the vault. Maybe they wanted the GECK?

Criminology 101, always investigate where the suspect lived.

The Father may have left notes and research about the Purifier at Vault 101. If I was investigating the technology, and I had no access to the Purifier or the team working on it, that is the first place I would look.