You're not being daft - it's a frightening condition, for your Ds and for you.
My DS1 had wheezy episodes from around his first birthday, usually in conjunction with a cold. He was given a blue puffer, and had several doses of the steroid tablets prednisolone. Then at 18 months he was given the brown preventer puffer, to take twice daily.
This didn't stop the attacks completely and he was hospitalised in January. Since then he's been on all the above, plus montelukast granules that seem to be keeping a lid on things - he's been a bit chesty and wheezy but we've been able to control it with the blue puffer.
As far as I understand it, they don't officially disgnose asthma in toddlers becuase to do so means doing a number of tests, like the peak flow tests, where you have to blow hard into a tube, and it's just not possible to get toddlers to do this effectively.
If the symptoms are there though, they will treat it as asthma, with preventer steroids etc.
If I were you I would keep taking him back to the doctor every time you feel that the blue puffer is not controlling the symptoms, or you ar having to use the blue puffer a lot, ie more than once or twice a week. if the blue puffer controls his symptoms then it's doing it's job.
As many have said before, kids often grow out of asthma, especially the sort that is triggered by cold viruses, so I wouldn't worry too much about sport just yet.
Virtually all my cousins had asthma as young children but most of them were fine by 7 or 8, and Im really hoping that will be the case for DS1. Ds2 is 4 months so we'll have to wait to see if he develops symptoms too. Fingers crossed he doesn't.