Premise: The question is proposing a somehow common joke in the MLP fandom. "What if the Pinkie Pie that was left in 'Too many Pinkie Pies' wasn't the original one?". Discussion about this meme aren't very common now, but were pretty common in the past. Giving a definitive answer is also pretty hard, since it is very easy to argue that the show didn't provide any direct evidence of the saved Pinkie being the original Pinkie. That said there isn't any proof of the opposite being true either, so by Occam's Razor principle I would just assume she is the same Pinke - unless you want to argue too that maybe every episode is set in a different but very similar parallel dimension.
Anyway, the question does also contain some false assumptions:
- The copies weren't identical to the original: even the first copy that was made didn't share the complete memory of the original Pinkie Pie. She didn't know where Applejack farm was, and when she meet Fluttershy on the road she didn't know her name (and she didn't remember it soon afterward while speaking to the original Pinkie either)
- The copies probably didn't degrade because of quantity but because "copy of copy": The first copy made was a copy of the original Pinkie. Then, after Fluttershy asks the copy to join her for a pic-nic two more copies are made. At this point, both Pinkies copy themselves, so we get another copy of the original Pinkie and a copy of a copy Pinkie. Things then start to go worse, since the copies start to make more copies until "Too Many Pinkie Pies"... So, it is very reasonable to say that the "degeneration" was caused by "copying the copies".
- The Cutie Marks on the copy didn't have to be real: it is unclear what the real nature of the copies was, but this doesn't mean their Marks have to be real Cutie Marks. For what we know, they can just be visual copies with no meaning. There are some examples of creatures in the show imitating the appearance of ponies - mostly changelings and Discord: both were perfectly able to visually reproduce a Cutie Mark without any problem.
- The copies weren't more "powerful" than the original Pinkie in any way: it is not like the copies had increased 4th wall breaking powers were compared to the original. As b_jonas already stated, by season 3 Pinkie had already given many displays of those same "powers" in multiple occasions.
- The ponies didn't actually send back all the copies: as show in one episode, at least one clone managed to somehow escape on that day. Her location is currently unknown, but at least once she was in the same bar Pinkie was, sitting on a nearby table. No one seemed to notice her.
- The copies were degraded copies of the original: now, this depends on how you define degraded. As the preceding point demonstrate, given enough time a copy would have been able to "grown up" in a more considerate pony.
- The pond doesn't just replicates ponies: again, a future episode S7 E24 “Uncommon Bond” shows that the pond can also replicate Maud's pet rock, Boulder.
So, given all of this premises, we can now try to understand the true nature of the clones. Again, this is just my personal interpretation of the story, but I consider it to be a fairly reasonable one.
The copies were actual copies of Pinkie Pie original appearance and the personality she was born with. What they actually lacked were all the experiences Pinkie had during her life. So, in a way, they were like newborn child, in an adult body, in a world that they didn't know. They went "rampaging" just because they wanted to have fun, in the only way they known.
Twilight idea was probably also based on that assumption too. Because the copies were just made, they also didn't had the time to develop strong felling for their friends and the world they lived in. So, when confronted with the "watch the paint dry" challenge, they all fail. Based on the "copies of copies" premise, one could also argue that the latest-made copies where those who were less attached to the world and thus the first to fail...
... which also means that probably the first copy made, the one that actually meet Fluttershy on her way to Sweet Acres and had a little more time to develop her feelings could be the one that escaped and is traveling across Equestria.
Another thing worth noticing is that the episode actually follows the original Pinkie Pie most of the time. We can see how she becomes progressively more depressed and worried as the episode goes by, somehow confirming that she is the only one among all the Pinkies that can really understand the situation and what is at stake. None of the other clones seems to care, they were just focused on having fun: again, that probably is a direct consequence of their lack of memory and thus inability to understand what the other mane six represent to them and what losing them would have meant. As expected, the "depressed Pinkie" is the one that in the end passes the test.
If we accept this - the depressed Pinkie is the same during all the episode - then it is logical consequence to think that she must be the original Pinkie too. The only way to escape that assumption is to argue that when the original Pinkie copied herself neither of the resulting two Pinkies were the original one.
While this is possible it would basically go against the main theme of the show - Friendship Is Magic. The point of the episode was to say that one should treasure the time he/she has to spend with friends, instead of keep wanting "more". If the original Pinkie had indeed disappeared in that episode... the lesson would become pretty empty.
Anyway, I think the whole argument can be closed by quoting an ancient book written by the Chinese philosopher 莊子 Zhūangzi:
Once upon a time, I, Chuang Chou, dreamt I was a butterfly, fluttering hither and thither, to all intents and purposes a butterfly. I was conscious only of my happiness as a butterfly, unaware that I was Chou. Soon I awaked, and there I was, veritably myself again. Now I do not know whether I was then a man dreaming I was a butterfly, or whether I am now a butterfly, dreaming I am a man.