The free tier shows that there are only three payments per month.

Is that the number of forms that can accept payments or the number of actual payments allowed? My form links to Stripe to setup a Subscription, but the payment doesn't actually take place: it's on a trial. Would this count as a payment?

It is the number of allowed payments. 

Payment Submissions - This is the number of submissions that have payment integrations on them. So for example, we have Paypal payment integration on our form, if a user fills out the form, submitted it and successfully completed the payment, then that would be counted as a payment submission. 

My form links to Stripe to setup a Subscription, but the payment doesn't actually take place: it's on a trial. Would this count as a payment?

Yes, that would count as a payment submission.