My understanding goes something like this:
.5 V/580 (12.7x81) = developed in conjunction with the military 1917-24, to meet a specification for a tri-service HMG round.
.5 V/565 (12.7x81SR) = private venture by Vickers for export purposes, reportedly to get around official objections to exporting the latest official ammo (i.e. the rimless .5V)
.5 Class D (12.7x120SR) = private venture by Vickers to address concerns that the .5V might not be powerful enough to meet customer requirements.
.661 Vickers (16.8x149SR) = RN specification to develop a much more powerful AA gun to replace the .5V (the Class D evidently wasn't enough)
15mm BESA (15x104) = off-the-shelf purchase by the army in c.1938 to acquire a light AFV gun which was substantially more powerful than the .5V, to obtain better penetration. I suspect that there might have been an element of "buy Czech" here, simply because the army had already adopted the Bren and they bought the 7.92mm BESA at the same time - they were obviously very impressed (and rightly so) by the Czech gun designers.