The thresholds are usually 1 to 2 inches wider than the door is thick, with a molded or beveled edge. The flat center portion goes under the door (duh) and the bevel is scribed to the door rabbet, and returned with a 40 or 45* angle if necessary.
A picture would be worth a thousand words, if only...
Casey

Hard maple flooring. The transitions are made from pieces of flooring and installed with trim screws. The screw heads are left showing as a design feature. The screw heads could be concealed with plugs.

Interior hall bath treatment--door closed{{!gwi}}

Door open--{{!gwi}}

Master bath, door closed---{{!gwi}}

Door open---{{!gwi}}

Entry hall to living room/center hall---{{!gwi}}

A "threshold" is for an exterior door and provides weather protection and drains to the outside.

For an interior door a raised "saddle" allows a small undercut on the door bottom and yet the necessary clearance for area rugs.

If undercut size or rug clearance is not an issue you would only need a "transition" or "reducer" strip between the different materials (or material heights) which should be narrow and located so it can't be seen from either side of the door if possible. At tile flooring in bathrooms it is normally made of marble with a beveled edge.

I think that what your contractor is referring to is a transitional molding, which is intended to make a reasonable transition between to distinct materials and/or heights.
Fit between the two materials or the down side routed on the back to slightly overlap to the lower material.
In a very short time, it will all but disappear.