If my mom put money into a bank account and put me as a joint account owner, is that considered a gift to me?

The equity from the house, half of that belongs to your Dad. Therefore when the time comes for Dad to go into a skilled nursing home, he will need to be self-pay. In the mean time he can sign up for Medicaid [which is different from Medicare and Social Security] to pay for his future care in the nursing home.

Please also note that if you pay anything to YOURSELF that is not supported by a caregiver contract that is signed with terms that are accepted by Medicaid, it MAY be considered a gift. I've known people that paid for gas, etc. and it was flagged by Medicaid because there was not a contract in place and it was considered something that a child would do for a parent....

It ought to be fine. Just be sure the account is listed under your mothers social security number and not yours. At least this should take care of the gift aspect - it would help you you were an actual POA with that notated on the checks after your printed name.

How the total amount in the account is counted as an asset that night put your father over the assets held limit might become an issue, however.