You should not ever have a delay in your ISR. All that will do is cause it to catch the first pulse but miss some subsequent ones. This is no help at all to solve your problem and just introduces another problem.
I understand you're saying that with delay set to zero the counter counts up even when there are no coins inserted, and it does this in all the interrupt modes.
You say you have checked and verified the wiring so I accept that what you're counting is the actual signal output from the coin detector.
What type of electrical output does the coin detector provide? I mean, is it outputting a digital pulse on a signal wire (if so, at what voltage and frequency?) or is it giving a loop disconnect type output? If it's a loop disconnect, you'd need a pull-up or pull-down to prevent a floating input which would cause symptoms just like you describe.
The other possibility is that the coin detector is triggering spuriously. Have you investigated the "Coin detection accuracy setting" options on the detector?