Does renting a room or apartment count as renting a home?

Do you mean that you are not the landlord, but instead that you are the one paying the rent?

I assume you are asking this question because your state taxes personal income and allows a "renter's credit" of some sort on your state tax return. The answer is yes if that room or apartment is your primary residence. Otherwise, no.

If you are asking in relation to a renter's credit on a state tax return, renting a room might not count. Check the rules for your particular state. I know of one state where, to qualify for the credit, the unit that you rent must have "its own separate kitchen and bathroom that you did not share with occupants of other units in the building."