IANAL (I Am Not A Lawyer)
A no contest plea or nolo contendere is like saying, "There is no way for me to contest this charge, as I have no evidence to the contrary, but I am not admitting guilt." The judge then makes a decision of whether or not to convict you based on the evidence presented against you, which is usually a verdict of guilty since there is no defense. If you consider yourself a felon or if you have a felony on your record, I'm pretty sure that means you were convicted of a felony. Doesn't matter what you plead, ATF Form 4473 sec. 11.c asks if you were convicted. Also as was already covered, if your sentence could have exceeded 1 year you are disqualified even if you are not a felon.
You could have been charged with a felony, but plead no contest to a lesser misdemeanor with a penalty of not more than 1 year and still be legit, but you'd still have been convicted of a misdemeanor.
Just because someone pleads not guilty does not mean they are not guilty and therefore not convicted, that's just the plea they entered. The judge or jury may still find the person guilty. By the same logic, if you are not a felon that would mean anyone who enters a not guilty plea on a felony is also not a felon - which makes no sense and is not true.
I'd say you've gotten especially lucky in passing background checks, but if you are indeed a felon you've committed several violations including Federal perjury. I'd probably delete this thread and contact a lawyer immediately. Don't want to get caught as a felon with illegal firearms.
Again, I am not a lawyer, and I could very well be wrong about this, but I don't think I am. I think the only advantages to a no contest plea are exhibited in a civil matter regarding the charge or just the fact that you never actually admitted to doing anything wrong.