A threshold is at an exterior door to stop weather penetration. A saddle is at an interior door to provide a larger door undercut for area rug clearance. A strip between two flooring materials is called a reducer strip wherever it occurs. The number of possibilities for the location and the difference between the two materials makes the question impossible to answer without making wild guesses

You dont have to take out the last row to install one instead of the schluter being installed under the tile you reverse it and install it under the carpet tight to the tile. ( you have to clean the edge of the tile ). Although you would need someone with a bit of expertise to fit the carpet properly. There are rubber or vinyl T moldings that you could use like you mentioned , They will work fine . However part of it will sit on top of the tile raising the profile and creating a tripping hazard . It will be a small hazard but never the less it is something that if you can avoid you should.Go to a local carpet or flooring supplier and they should be able to sell you one and as you say just glue it down cut the carpet to go under the lip of the molding.the carpet should be stretched and held into place by a piece of gripper by the molding so it wont wrinkle there. Please use a decent glue or you will end up just kicking it off. Normally you would not put a cut tile in front of a door like that. Sounds like you had a couple of problems with those guys , sorry to hear that. Good luck

Never assume a flooring installer will finish the transition to another kind of flooring. A tile-setter does not carry carpeting tools. Flooring installers don't trim the bottom of doors. These disclaimers are usually stated in fine print at the bottom of the proposal. I would have completed the transition, and trimmed the bottom of the door. I make sure that part is clearly stated in the proposal, and how much I'm charging for that. Usually 50 to $100 per transition, depending on how heavy the door is to take off. For this, I would use the thin metal floor reducer. Here are the anchors best suited for this application: Here is a link that might be useful: Hilti HPS light duty concrete anchors

I've always seen the carpet tucked into the tile/carpet transition using the same scraping/tucking tool that's used at the baseboards. I guess it's possible to do this poorly and end up with fraying at the carpet edge, but I've had many like this in my homes over the years and none have frayed. This doesn't typically work at vinyl/carpet transitions because there's not enough height on the vinyl side of the transition to tuck the carpet edge in, which is why transition pieces are almost always indicated at vinyl/carpet boundaries. But carpet/tile boundaries can be finished looking and durable without transition pieces or binding the carpet edge. This post was edited by GauchoGordo1993 on Wed, Dec 4, 13 at 19:48