Im not an attorney but i do write contracts for my business, and I'd advise being careful of the phrase "any rental agreement that doesn't have a specific lease term"....in the noted caseload above...

If the landlord is a tool, and If he has the ability to bring you to say small claims court, you do at least have an implied oral agreement...meaning: you have for a period of time paid a set amount of rent to the landlord that you both agreed to at some point.

I'd think at most you could be on the hook for that last month, but not the increase. Again, if he brings you to court...

Good luck!