My opinion on how you can log training time, as I understand your questions, is noted below.
I believe that 14 CFR 61.51 and 14 CFR 61.63 answer all of your questions.
The type of aircraft you are training in (Cessna 210, Cessna 310, HondaJet HA-420 single pilot jet, or B737, etc.) would not change the way that "training time" would be logged. It would be logged (in accordance with 61.51) as "training time" received by you as long as the instructor was appropriately qualified.
I cannot locate any regulation that would indicate that you would not be able to log this as "training time" received in the category and class simply because the aircraft requires a "type rating" to fly as PIC.
Can you log dual received in a single-pilot turbojet on Part 91 flights if the PIC is type-rated, holds a valid MEI and instruction is being given?
You can log "training time" (dual received) in the type of aircraft you received the training in (e.g. HondaJet HA-420, a single pilot jet).
Can the AMEL-rated student log AMEL category and class time if receiving flight instruction in this scenario without holding a type rating in the aircraft, or can only dual received be logged. In the latter case could you log the time at all if it cannot be tied to a category and class.
FAR 61.51(b)(1)(iv) requires you to enter the "type" of aircraft that you are flying in your logbook. In the example aircraft you would enter the "type," which is a HondaJet HA-420 ("airplane" category, "multi-engine land" class), and the amount of training time received. It's automatically tied to the category and class because you have to enter its "type."
Similar scenarios that come to mind would be instruction for an initial type rating in the aircraft instead of a simulator. How is that time logged and what qualifications would that instructor and student need?
For an initial type rating in the example aircraft (HondaJet HA-420) the training time would be logged as noted above, and the student would need to hold at least a Private Pilot Certificate in any category and class. See FAR 61.63 and note how to add an additional category or class rating to an existing pilot certificate (if necessary) and the requirement to hold or concurrently obtain an instrument rating.
The instructor needs to be qualified in the same way as you note in the first paragraph of your question.
