Two answers to your question:
I think your recollection of that specific episode are wrong; I just re-watched the episode online and the "we came first" part was not there; after he says "you look Time Lord" he stares at her for a short period of uncomfortable silence/sexual tension before moving on.
However -- yes, there are different versions of the episodes, especially if you're watching them outside of the UK. I can only speak first-hand for the BBC America network but I suspect other foreign networks work the same.
The episodes are originally produced for the BBC, which is a publicly-funded network and thus has much fewer commercials; and usually only between episodes. (It's been a long time since I actually watched a program live on BBC but recent shows on YouTube would indicate that this is still how things work.) The run-time of the episodes can vary by several minutes, and the BBC just adjusts their commercial schedule to match.
When re-cut for air in the US, they have to fit within the typical 49-minutes-plus-commercials format, which means both editing for time, and figuring out where to put the breaks.
The most notable example of this I can remember is "The Eleventh Hour", the first Matt Smith episode. Uncut, with commercials added, it runs 1:10 or so, but as it was a special premier episode, BBCA ran the entire episode. However, subsequent episodes are edited, cutting scenes like the "Amy cooks for The Doctor" scene and "The Doctor changes clothes" scene down to almost nothing. A similar thing happened with the Tenth Doctor's first episode; when that episode is rebroadcast on BBCA they cut out this entire scene, including my favorite line: "Am I... ginger?"
On the 'net, you are most likely watching an episode that was either ripped from the BBC online player or from the DVDs, and in both cases you'll be seeing the full episode. When you watch on TV, however, you may end up seeing an edited version with bits cut out to make things fit their schedule.