Our ECAM can't measure consecutive start attempts.
Our limitations: 2 mins on, 15 secs off, 2 mins on, 15 secs off, 1 min on 30 mins off. So in the unlikely event you get to the third start, does you ECAM 'Know' that this is your third start attempt? We have pretty old -320's so maybe the newer ones are smarter.
Engine warm up for our engines is measured from 'at or near idle.' There fore we hack the clock when the starter valve opens then re hack after the second engine is at or very nearly stabilized at idle.
So many versions of a very good airplane flying around, what I just described is how we operate in our little world....