Conviction is what happens in the court, from your description you’ve had no convictions at all yet.
You should be offered an awareness course (if the offence the previous one was offered for was more than 3 years before these) as well as two fixed penalties.
You may have a case that the two 4 minutes apart amount to a single continuous offence and only get punished for one, some forces are better than others at this but it may need pointing out to them.
My concern is if you picked up 3 in a day, how many more may arrive now?