Two documents are doing two different jobs. The note is your promise to repay. The security instrument (called a mortgage or a deed of trust depending on your state) is what pledges the house as collateral, and that's where the occupancy covenant lives. The certification you signed at closing is a statement about your intent at that moment.
You read it correctly that the covenant language typically runs from the date you occupy rather than the date you close, and that's why the two dates matter separately. Programs also layer their own conditions on top, and those get updated, so the only version that governs your loan is the language in your own documents. Ask your lender to point at the clause and confirm in writing how they read it.
On what happens if someone leaves early: breaching an occupancy covenant gives the lender contract remedies, which can include calling the loan due. The more serious exposure is when someone signed an intent-to-occupy certification they never meant, because a false statement to get a federally related mortgage is a different category of problem entirely. That one is a question for a lawyer, not a forum.
The piece people miss is what happens after the period ends. You generally don't need lender permission to rent it out at that point, and the loan terms don't change. What does change is your insurance, since a homeowner's policy assumes you live there and a landlord policy is a different form. Also check the HOA documents if there are any, because rental caps and minimum lease terms can block the conversion even when your lender has no objection.
