Can I rent out my primary residence?

Take some of that 10 grand and spend it on an attorney who can tell you definitively.

You can own property anywhere you want.

It's what state you declare you are a resident of that matters.

Talk to a real estate lawyer to make sure. There are tax implications, for sure.

If you move out, and rent it to someone else, it isn't really your primary residence anymore, is it?

Yeah, a lawyer can probably tell you for sure.

But I am guessing that if you move out and rent the place there is no way in hell the city is going to see that as you living there which means you will have violated the agreement and will have to pay back the money.

Some cities it's illegal to sublet ones apartment. As Jespah said already... hire a real estate specialized lawyer and use his expertise rather then a strangers who can't ethically give you legal advice.

What about your Florida Homesteading exemption?
That $50,000 knocked off the value of your home for property tax purposes might be a cool chunk of change come tax time.

"Matters" for what? His question is whether or not his moving to Philadelphia will violate his contract with Jacksonville for the house to remain his "primary residence" for the next 2.5 years. It's doubtful that merely "declaring" his domicile to be Florida will result in a finding that his Jacksonville home is his "primary residence," when he is in fact primarily living in a different house in a different state.

But what is the definition of "primary residence"?

We can all guess at the answer, but really that is the question that should be posed to the attorney that Jespah suggested the OP consult.