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

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.

In your situation and having a very thick cased opening I believe a wide marble saddle would look good. Transition materials such as saddles or Schluter Schiene metals protect the edge of the tile from chipping. You do need to leave a slight expansion no matter which way you go. I have several photos showing a color matched grout caulk in the expansion gap. Another look to consider is stop the tile on side, stop the wood on the opposite side and fill the 12" area with hardwood ran perpendicular.... called perpendicular skirting or perp skirt. Here is a link that might be useful: Wood to tile transitions

You've had a Schluter edge put on a tile that needed a proper CARPET REDUCER. A REDUCER is just that. It REDUCES the height of the tile DOWN TO the height of the carpet. That's why they are called reducers. They are wedge-looking pieces that make this a non-issue. Your carpet/pad combination is not thick enough to handle the "L" you have. You will need to have a reducer added. It won't be pretty. The schluter "L" has been installed at the same time as the tile...which means it is sitting UNDERNEATH the tile. It will take some effort to change this. A decent tiler will handle this without much issue. Whomsoever made the original decision will pay for the change. That's how these types of change orders work in new builds. The reducer install will be much cheaper than removing the carpet and replacing the pad and then reinstalling it all.