"Bitcoin-XT" is another implementation of the bitcoin protocol, started by Mike Hearn. It is still compatible with Bitcoin-core; either version can be used with the same wallet files. Gavin recently said that he would be joining Mike in the development of this software. Gavin mentioned very recently that he may use this version to push for the 20 MB change, since the current core devs do not seem to be in a hurry,
Bitcoin-XT is not a re-implementation, it's a patch on top of Bitcoin Core that adds two features, double-spend relaying (which can be flagged for inspection) and the BIP64 getutxos message (which he needs for Lighthouse). The reason XT exists is because Mike Hearn can't get BIP64 implementation merged into Core (
https://github.com/bitcoin/bitcoin/pull/4351). Gavin happened to be for putting BIP 64 in, so moving forward if this happened, Bitcoin-XT would be a proper fork based on the actual Core codebase, simply with these two differences plus the blocksize changes. The fact that moving to XT is considered a viable option might suggest that there is a much larger "turf war" going on here since at least the middle of 2014.