I agree with you. ZU-23-2 uses a gun 2А14 with air cooling. The same gun with water cooling 2А7 for ZSU-23-4
The gun 2А14 is used at modernisation BTR-70 in Armenia
Based on what I've been able to see from available photos, the 23mm 2A7 cannon used to upgrade the BRDM-2 is the standard water cooled one with coolant plumbing and all, while the BTR-70, although having the same basic 2A7 cannon, has all water cooling accessories removed. Probably because as an AFV weapon it will not need to fire long sustained bursts, but just short bursts or single shots, which makes water cooling somewhat superfluous.