Someone who corrects people using improper grammar is not a Grammar Nazi.
Someone who goes around YouTube, Facebook, Tumblr and more with posts consisting only of rants about improper grammar or corrections is a Grammar Nazi.
The second person would definitely be a grammar Nazi, but the first is unclear. Do you mean:
Uses incorrect grammar to correct someone else's incorrect grammar? That would not be grammar Nazi-ism. That would be hypocrisy.
OR do you mean:
Corrects someone else's incorrect grammar without intentionally using incorrect grammar? If someone does this often enough, that is grammar Nazi-ism. Grammar Nazi-ism is determined by the frequency of corrections, so doing it once only doesn't make once a grammar Nazi. However, if this is repeated over and over again, the grammar Nazi-ism increases with each repetition.
They're was no point in my post where I used the wrong, "They're", "There" or "Their".
You may not have intentionally made an error, but you made the same error in this post and the previous. The bolded word in the above and below quotes should be "There".
Seriously, if you go over they're to another forum you will see what I mean when the whole forum is filled with "*They're" and you see 10 people that post within about a minute about the small errors in someones post.
As explained earlier in this post, that correction alone does not make me a grammar Nazi. I merely made the correction to either expose trolling or to make sure you don't make the same mistake again.