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

It sounds like you have several issues: A) Squeeks B) Cold Floors C) Concern about floor level differences Regarding A, the builder is basically correct - 3/4 OSB is a building standard now. It is tongue and groove and glued down and together. Over time is will begin to squeek. You know where the squeeks are and when removing existing flooring, nail the subfloor tight again. Should eliminate, or at least reduce the squeek. Regarding B, See if you can have foam insulation injected into the cantilevered space. As to heating, it still may be necessary but why not minimize its burden by controlling the root cause. Regarding C, Several things here. Your current vinyl should be installed on a thin underlayment. Combined they are about 1/4" thick. It will be necessary to remove these. (This will be the time to inject foam insulation) A tile floor needs 1.25" minimum structural support which means a 1/2" thickness of concrete board will be cemented and screwed atop the existing subfloor. Then your heating mat followed by tile. So, you'll be gaining about 3/4" above the current level. Your tile shop will have a marble threshold with a beveled edge to minimize this effect. We have this same situation is our home and it is negligable. You may or may not have to trim the door. Base trim would be removed in the bath to do this work. It can then be reinstalled, replaced or have the tile setter install tile as base trim - that's what we did. Here is a link to an updated bath we did and amongst the photos you can see the threshold and tile base trim. Hope this helps. Good luck. Here is a link that might be useful: Bathroom

I'm the OP. Those marble strips are nice but they only come in white. Since our floor is dark they wouldn't work for us. We ended up going with wood. There was no molding of the right shape or height, especially since we were trying to bridge a height transition: the tile being higher. We bought ranch molding that was designed for window and door frames. If you are not familiar with ranch molding, in profile it has a slope: one side is high and the other side is lower. The installers has to shave some wood off the back to make it fit (to reduce the height), but that was their problem since they had promised they would come up with something nice for us and had not. We had them bring the wood right before the weekend and then we stained and varnished it to match our other moldings. On Monday they came back and installed it. The carpeting is a berber, and the edge is tucked lightly under the lowest edge. The high side butts against the tile/durock and a small caulk line is in between.

Many thanks :) - This is our basement. My daughter is moving back home after 6 years in college (have never experienced what an "empty nest" feels like - LOL ) - We are fixing up the basement for her - She is afraid that all tiles would be very cold. The entire basement has carpet for the last 24 years but the water came in through the windows during the last big storm so now we try to replace the "potentially can get wet" area with titles (about 1/5 of the entire basement). We try to make it look artistic .. hence the curve comes in the picture. " Because it's a berber carpet you may want to consider a vinyl snap in. " Will go to the local professional flooring store(s) and see if they have this. "The best bet would probably be to find a carpet workshop, cut the curve in your carpet, bring it to a pro and have them bind the edge. Install the pieces of tack strip right next to the tile, run the pad up to it and tack the carpet. You might even want to do the carpet first, considering how hard it could be to get it stretched perfectly to the right spot. Then tape the edge and cover the carpet while the tile is installed and grouted." Great suggestion - I will look into this. Any advices on the grout width and any other ideas with this curve thing are greatly appreciated.