How to reconnect a relationship with a professor Today

I'm currently in my final MSc year, and I have finished working for my thesis. I am under supervision of professor X, which I find myself at ease and actually enjoy what I do. My field is Chemical Engineering.

Background

Before starting my thesis, during the "thesis choice" period, I was contacted by professor Y, which I was in very good relations (we did some external projects before the thesis, with good results). One of the major aspects of working with Y was the possibility to go abroad, as he sent out his students quite often for thesis work. The drawback was the lack of direct supervision. Professor X on the other hand contacted me, but he was not in favour of the abroad experience, since he only works as a direct supervisor with thesis students.

Issue (first mistake)

I had two initial proposals from both professors. At first Y's proposal was way more interesting than X. I got a week to confirm Y, but after the third day, Y's initial proposal was changed (in a uninteresting project). 

I thus contacted X, and after a week arranged for a thesis meeting. X's offer was also changed, but now very being intriguing, and Y remaining proposals were not as interesting as X. I said yes to prof. X and went personally to Y's office to tell my decision. 

When I entered Y's office (without notice, as MSc students - per Y rule - are free to pass by his office freely), he welcomed me with a project that "was sure I would love", and there were also external profs and industry reps (for which I had not been notified) and quickly explained to me the thesis, all taking like I was in from the start. This "meeting" with me was not programmed, as I expected only Y being in his office.

Right after this meeting I made a big mistake. From all that was said in the meeting, and the confusion that was created, I said (verbally) yes to Y.

Issue's Aftermath (second mistake)

I continued working with X, and received NO news or advices or any message from Y. My suspect is that Y's project came to nothing, but seeing that no messages came from Y, I made the second mistake as I continued working with X not alerting Y. I know that a single e-mail could have resolved the issue, but since that period was also characterized by a grave family issue, that e-mail was never sent.

Question

As the months passed and considering my today situation, and (fortunately) all the personal issues have subsided, what could be the best course of action of regaining contact with Y? After all, he did nothing wrong to me and I know I made some terrible mistakes, I want to apologize such to not truncate our 4 year relationship.
I think you have gotten yourself into a pretty deep mess if it is really as bad as you suggest and not just a sense you have of the situation. But, the only way to proceed is to go visit the professor in person and steel yourself for whatever occurs. I'd make an appointment first, to give the other person a chance to think about their own response. 

You may well owe an apology, so prepare to give it. But, as long as the prof isn't a monster, it may well turn out better than you fear. But it may also take a while to completely dispel any bad feelings if they exist so a single meeting may not turn things around like flipping a switch. But a face to face meeting will give both of you a sense whether a continued professional relationship is possible and desirable. 

It may be the professor was disappointed when you left, but also possible that he/she is so busy generally that it didn't register too much. 

The past is what it is and can't be changed. The future is up you you. But expect that the transition may be uncomfortable.