Does it stand for something different, though?

It doesn't stand for Tsu.
doubles the consonant of the character after it.

Ex:
kekkon けっこん,
kippu きっぷ
kitte きって
kissaten　きっさてん

So basically if there is a っ before こ, it will make it kko or kka for　か, so on and so forth.

It has no inherent pronunciation of its own (OK, technically it does, but just not in the way I'm talking about). It just looks like つ. Observe the difference: つっ.

It's like how e and ə are different even though they're the same shape, just with a different geometric transformation applied to it. The same with how – and — and - are all three different dashes with different meanings. One is an en dash, one an em dash, and one a hyphen.

This is very nearly correct. See my post and duo797's earlier, discussing what it really does. A decent approximation for beginners is that it means "double the next consonant." That's not entirely correct, and once you've started reading intermediate-level dialog, you'll see things like
あっオマエだ！ Which is "A-- omae da!" not "Aoomae da" or something.

Excellent comment, KyleGoetz.

As a native speaker of Japanese who also happens to speak decent English, I have a big issue with the use of the term "double consonants" with regards to the っ for the following reasons.

1. It can mislead the beginning Japanese student to fall into thinking that the same sound exists in English as double consonants do exist in English as well. The truth is the っ sound exists only in several languages in the whole world such as Japanese, Italian, Arabic, Russian. English ISN'T one of those languages. If you thought it was, you've been mispronouncing the っ.

2. The term "double consonants" is completely irrelevant in the following cases. Native speakers have no trouble whatsoever in producing the っ sound at the very end of a word where no consonant or any other sound follows the っ. This is very common in casual speech.

We can say:

はやっ How fast!
いたっ Ouch!
これだっ This is it!
みるなっ Don't look!

I'm still a little confused but now it became clearer to me somewhat... :)

ありがとうね...

I'm currently learning Hiragana... :o

Strangely, as much as I've seen this, I've never actually looked these up.

Would it be correct to say that in these examples, the っ makes the word end more abruptly? Almost as if you hold your breath in the middle of the last sound. Sorta hard to explain...

It's hard for me to explain, too, but "abruptly" seems to capture the feeling.
You sort of hold your breath for a fraction of a second and then quickly let it explode.

The majority of the non-Japanese I talk to in my area of Tokyo are English- and Chinese-speakers. When they produce this っ sound, well over half of them fail and end up just elongating the preceding syllable.

For those out there who want to figure out what Nagoyankee and I and duo797 are talking about, try this:
1. start saying "hee hee hee hee hee" with no gaps between the "hee"s
2. now try to abruptly stop a "hee" and have the "hee"s become separate sounds instead of flowing into each other
3. that pause is the same thing as a っ.

I attempted to explain in this video, but I wouldn't be surprised if my accent is heavily gaijin. I was more focused on explaining the っ. I know, academically, what it should sound like. I hope I pulled it off, but we'll find out if some native speaker wishes to pwn me to Andromeda.

YouTube - little tsu explanation

Going to subscribe, than you!