I agree. Different tests have different levels of sensitivity. I have even had a positive on FR, then a negative at the Dr.'s urine test. FR has insanely low sensitivity levels, while my Dr.s didn't register until levels hit 100 (as I found out later). I'm not certain when you took the generic brand, but now that you are 3 days late, almost every test should register positive. Make sure you are using a pink dye, first morning urine and follow directions closely. Or, as Scout said, see your Dr. for bloodwork. All the best to you!
I think First Response tests are pretty accurate and they are really sensitive too. If you had a faint line then that would indicate pregnancy. You can ask your ob to take blood and test for hCG, which is what will confirm if you are pregnant or not. You could also wait a couple days and test again with another First Response and see what you get. Honestly, if I got a faint line on the FR, I'd assume I was pregnant and call my ob to come in for blood work to confirm.
Good luck! I sure hope it's a BFP!! (and really sounds like it is to me)