"TS" = transport stream. Only a few select formats can be used as transport streams, namely MPEG-1, MPEG-2 and some of the MPEG-4 variations (like H.264). Transport streams were originally created for broadcasting, but its use was expanded for post-DVD disc and camera formats. Part of that expansion was the inclusion of extra timecode data, which accounts for the additional few bits per stream segment.
Possible file extensions include:
The "m2" portion of that last extension reflects that the stream is MPEG-2, at least historically. The MPEG-2 TS version transport container is also used for MPEG-4 AVC/H.264 content on Blu-ray media.
I'm aware of most of this kind of content, but don't deal with it extensively. A lot of video work is still overwhelmingly based around standard definition content (DV, DVD, DVB). Transport streams are also distribution formats, so most handling of these files are handled by end-users (viewers, consumers, home enthusiasts), instead of the earlier generation Avid, ProRes, MXF and HDV type streams and containers -- which is where I do a lot of work.
For the most part, .ts and .m2ts are the same thing, though expressed in slightly different ways.
Playback problems are more likely a reflection of the player being picky, instead of the files being bad in any way. That's always been common with "added on" type playback features. It used to be confined to DVD players and DVD recorders, but Blu-ray and current generation HDTVs are playing their hand at format expansion. In some ways, that's nice -- but it's also bad, because there's no standardized way to support the many formats that exist.
Enter the "media center". Have you consider simply using a top-of-the-line media player, like the excellent Western Digital WDTV units? I used to be frustrated on a weekly basis trying to play random files on a DVD player or hacked XBOX, as nothing ever seemed to work as desired. The WDTV really changed all that -- it's rare that the WDTV won't properly play a common file.
I urge you to consider a WDTV model:
http://www.amazon.com/gp/redirect.ht...reative=390957
It will take the nuisance out of what should be an enjoyable experience (watching video).
For M2TS support, you want the WDTV Live Plus model:
http://www.amazon.com/gp/product/B00...SIN=B003MVZ60I
As Joekster points out, trying to guess at the specific niche compatible with the HDTV will be a source of continued frustration.