Life after graduation; should I update my academic reference on my career progression?
Do professors empirically appreciate their former students keeping them in the loop of their career progress? 


You know your professor's personalities better than anyone here on SE. If they seemed cold and remote to you while you worked with them, then you probably won't gain much by keeping in touch. But it seems you have 'a rather a close relationship' with them. If that is the case, then I would say: Yes, get in contact with them.

Understand, you aren't just 'keeping in touch'. What you are actually doing is professional networking. If these professors work in your field, or have any similarities, then it is to your advantage to stay in touch with people in your field. At some point, you may find that they can help you with future employment. Or perhaps as your own career advances, they will come to you for help in something. This is part of the 'game' of being successful in the professional world.

Tip: If you haven't seen these professors in a while, ask if you can stop by. Just try to have lunch together. Tell them about your new role. If you learn that someone did help you by being a reference, see to it to send that person a Thank You card. It's all part of the networking game.

Lastly, Now you haven't tagged what country your are in. If it is the case that in your particular country, in your particular culture/society, people don't network like this, then ignore my answer.