If I was added to an existing mortgage through a refinance, am I no longer considered a first time buyer?
The "true" answer is no, you're not. The practical answer is that a lender (or state agency) who wants to verify whether someone is a "first time homebuyer" will look at a couple of things: tax returns for mortgage deductions (which you may or may not have taken) and credit reports for mortgages.. Presumably your mortgage shows on your credit report, which makes it problematic to claim you've never owned a home. Hope that helps, let me know if you have any other questions.
You're not technically considered a first time home buyer but you can still purchase with an FHA loan (for owner occupied) or get a fannie mae loan. It would be tricky if the loan you are on now is FHA and you want another FHA loan (because of the owner-occupied rule) but other than that, there's just a few programs that have real "first time home buyer" benefits. So,it's not like you're really missing out on a lot.
The issue now is that you are obligated on another mortgage...so it will be included in your liablilties (the entire PITI payment) and you will have to qualify with it (and your new mortgage payment) to get the new loan.
The good news is that you can become a "first time home buyer" over and over again. If you haven't had an interest in a home in the last 3 years, you're considered a "first time buyer".