Your Doc should Test FT4, FT3, and TSH.
She may not have heard of these tests, but the lab will have.
If she is so out of date, she needs to research. If she won't, or can't I think you may need a new dr.
No they are not the same. Total T3 is just what it sounds like---all the T3 in your blood. Most of this T3 is inactive because it is bound up with protein molecules. Only the small portion that is not bound with protein (thus free) is active in regulating metabolism and all the other body functions it affects. That is why it is important to know the free T3 level, not total T3.
I'm not too optimistic about your doctor if she has never heard of FT3. Your only hope will be if she is at least willing to test and adjust FT3 and FT4 levels with whatever type of meds are required to relieve your symptoms, without being constrained by resultant TSH levels. If not then you will need to find a good thyroid doctor that will do so.