OK, here’s my take, as an ‘evidence based’ chiro student and avid lifter who has also had shoulder rehab issues.
Firstly, without a proper examination, you will never know what exactly is clicking. BUT, there is obviously some form of instability/early degeneration there, that was not there before. Not good.
It could be long head of biceps tendon, it could be a subacromial bone spur (not likely), AC joint instability or GH joint instability (quite possible).
Now, my recommendation would be to drop the behind neck press. It is not a sesnsible exercise choice for anyone, let alone someone who wants to maintain healthy shoulders. Why? because you have the joint at maximal end range, in a tight-packed position, where various structures that should only be approximated once in a blue moon, are being forced together and ground against each other with weight applied.
Anyway, besides stopping the BN press, you should look at incorporating some sound rehab exercises into your programme, to strengthen the rotator cuff muscles and help centrate the head of the humerus in the GH joint.
Here is a link to what I consider to be a good shoulder rehab/prehab routine:
I totally agree with the dropping behind the head pressing. If you do behind the head pull downs you should drop those to. In addition to what Bushy said you should also include a lot of face pulls. They will help stabilize the shoulder. Start with 10-12 reps.