What's the difference between these plugs?
Be careful -- receptacle type usually reflects voltage and phasing

First, the "250V" rating relates to socket insulation limits.  It has nothing to do with expected/conventional voltage on them.  (You can't exceed it obviously). 

The CS6365 is for split-phase 120/240V power, and its NEMA equivalent is NEMA 14.  

The CS8365 is for 3-phase delta 208V, its NEMA equivalent is NEMA 15. 

That is plainly stated in their catalog (if you know what to look for). 

I leave it to you to look up the bolded phrases. 

By the way... all 208V installations are actually "wye" (120V line to neutral), and can be used either wye or delta.  It is quite common for 120/240V split-phase appliances to be used in 120/208V 3-phase territory.  They simply attach 2 phases and 1 neutral, so the 120V sections work normally, and the 240V sections get sqrt(0.75) less voltage than they are used to. Resistive loads can live with this, they simply work at 3/4 power.  Motor loads, you'll have to check the nameplate to see what they can accept. However, the important part is these 120/208V 2-of-3-phase appliances are wired up NEMA 14 - same as 120/240V split-phase. 

But why on earth do some have a different connector?

The worrisome thing is that some of your machines have the different connector.  Usually, that means this machine requires a different voltage.  It's like if you go to plug in your RV and it has a TT30 type connector, that always means it is a 120V/30A RV.  An RV with a NEMA 14-50 means it wants 120/240V split-phase. 

So on these machines that have CS6365 -- look very, very closely at their labeling, nameplates and the like. In fact, post them here. You need to assume the CS6365 is correct and the machine is 120/240V split-phase (or 120/208V) requiring neutral, until clear evidence on the unit proves otherwise.  Don't fall to wishful thinking. 

I suspect that converting these to 208V delta would be a mistake.  You must assume they have sections which require 120V line to neutral, won't like 208V there, expect 240V line to line, and may be able to tolerate 208V line to line.  If that's the case, then you need to provision sockets with this H-H-N-G wiring.