In general you can glaze the profiles only or glaze the entire door. Sometimes it's referred to as dry or wet glazing, respectively. Glazing the whole door will tint the tone of the wood. Glazing just the profiles won't.

Tung oil(not really just tung oil, but a recipe of polymers as a rule) is an oil based finish, so it creates that amber tint. Tung oil finishes also need many more coats to provide the same protection as 2-3 coats of polyurethane finishes. And drying times for tung oil finishes are usually much longer per coat than poly. Pretty good descriptions

At this point, it seems like a lot of work to remove the threshold board and redo the floor. To really look like the same floor, you would need to remove quite a few boards of the existing floor to weave in new wood of random length. All the seams between the boards should be several inches from the adjacent boards so at least 2/3 of the existing boards would have to be modified. Then you would have to hope the new boards were an even size match for the existing which isn't guaranteed. Yes, if you measure. them with a tape measure, they will both look to be 2 and a quarter inches, but measure a 32 board (6ft) section on each side and you may find a small difference, just enough to be noticeable to your eye. Also mixing brands and ages of wood flooring can be like trying to mix Legos and Megablocks...they are compatible, not identical. It sounds like the installer is willing to work with you which is a positive. I suggest trying to get a written agreement with him that states he will permit you to have an extended period of time for rework. Then live with the floor as is for several months before you decide if you want the existing floor refinished to get a closer color to the new portion. You might find the color bothers you more as time passes but perhaps you will get used to it and not notice it as you start to live in your space

The cheap way: Remove carpet, add a decorative TRIM at the doorways (where the two woods meet) and lay the new wood PERPENDICULAR to the hardwood in the rest of the home. Sand and refinish all. The most APPROPRIATE way (to lay the new wood parallel to existing): Remove old wood floor, lay new plywood subfloor and lay new wood flooring parallel (in the same direction) as the existing hardwood. Sand and refinish all. You will need to choose between FINANCIAL vs. Aesthetic concerns. I'm someone who appreciates the TECHNICAL side of the building industry (aka. Best Practices) who is also a cheap/lazy person who pays attention to finances MUCH more than I pay attention to aesthetics. I choose the 'decorative trim' with the new wood running perpendicular to the existing wood. It is inside the budget AND it gets you the 'right' type of install (which you have already been told is best) AND it gets a solid hardwood throughout the house. Done. Done. And done. There is ONE option that NO ONE has discussed with you: Remove ALL flooring surfaces and install ONE wood floor. Sand and refinish.