For details on Azimuth, check out this post from 2017. There is a lovely explanation which clearly explains the concepts.
In my opinion, the difference between a 2-head and a 3-head deck is simply in the number of heads used. A 2-head deck uses a common head for both recording and playback (reproduction) while a 3 head deck has dedicated heads for recording and reproduction. The 3-head concept has been adopted from reel to reel deck technology. The head which is dedicated to reproduction or playback is called the repo-head.
Both deck types also have a separate erase head which is activated during recording.
3-head decks are generally more expensive and may have more bells and whistles that may be or may not be of use to the user, depending on the requirements.
The advantage of having a 3-head deck is that you can monitor what is actually being recorded on the tape in real time by using the tape monitor feature on the amplifier, facilitating what is called confidence monitoring. Where as in a 2-head deck, even though you can monitor the line feed that is being used for recording using the monitor feature on the amplifier, the actual recording on the tape cannot be monitored while recording and can only be one so after completing the recording and playing back the tape.
Another advantage of the 3-head deck is that the recording stages can be easily calibrated as one can monitor was is being recorded on the tape in real time but this feature is only useful from a technician's point of view and has no impact on the user.
Finally and again in my personal opinion, in audio gear, generally you will find dedicated components that are supposed to perform only a single function and combo components that can do many things together. People usually prefer dedicated components as if only one function is performed, resources are not shared and hence, performance is better. The same principle applies the 3-head decks. In this case, there is a dedicated head for recording and a dedicated head for reproduction, hence head calibration, matching with circuitry and overall performance can be maximised separately for recording and reproduction. Hence, if you look closely at spec-sheets of 3-head decks, they tend to have better performance specs when compared to 2-head decks.
Having said this, 2-head decks are not inferior by any means. There are many 2-head decks out there with similar specs and good performance and can out-perform 3-head decks. My personal experience with Nakamichi DR3 is that it performs much better than most 3-head decks. Also from a servicing perspective, it is easier to get a variety of head replacements for 2-head decks as this deck-type was most common. Getting replacement heads for 3-head decks can be a tricky and expensive proposition.