My ideas on what make a Final Fantasy a Final Fantasy:

1. Amazing music scores. Nobuo Uematsu was a genius with high quality music. Final Fantasy IX has the best music of any game I've ever seen, and then X comes close, with VIII and VII right behind. The earlier stuff is really good, too, but not quite my taste as much. XII, XIII, X-2, and XIII-2 had good soundtracks, but not the GREAT of the other 4 I mentioned.

2. A (compelling) story that actually makes sense. Again, Final Fantasy IX, X, and VII come to mind. VIII, too, until you get to Edea's assassination, and then the entire story after that is crap. IV and V were great, too. Hell, even if the story for I wasn't compelling, it made sense, unlike XIII and XII. XIII-2 made a bit more sense than XIII, and was easier to follow, but it still had some gray areas that just make it feel strange.

3. Great characters that are intertwined with the plot. Every non-optional character in Final Fantasy I-IX is necessary for the plot except for Edward in IV. If you removed any character, the plot would either be changed drastically, wouldn't work at all, or would have a VERY hard time staying on its original path. This is untrue of X, XII, and XIII, though. I found Tidus, Vaan, Penelo, Hope, Snow, and Sazh to be completely unnecessary in their own games. Remove any of them, and sure the plot will change a little, but it will still be pretty close, and will still work(if we're assuming XII and XIII's plots actually do work).

4. Real towns and areas to explore, and not just to run through. I-IX and XII had these in abundance. X, X-2, and XIII-2 had some areas that you could really explore. XIII had none. However, I still don't really count XII in that because though you could explore all the areas, there was never once a single thing to actually find, unless you stumbled upon a rare drop/spawn out of pure luck.

5. Replayability, and being able to play with more than 1 viable strategy. I-IX were great with this, and X was pretty good with it, even if your characters were pretty specialized until pretty late in the game. XII had more of this than any other game, whereas XIII, XIII-2, and X-2 you barely had any choice in a lot of encounters. There was one specific way to beat things, and if you didn't do it, you lost.

So, overall, the only ones I think contain all of these elements are I-IX, excluding VIII. Those are what I consider Final Fantasies in my mind, with X being pretty close to the definition. I think that's why I haven't like the new ones as "Final Fantasies", but instead just liked them as "JRPGs". They don't capture the old magic, sure, but I still think they're all great games. Just not Final Fantasy worthy like the old ones.