Polka, I don't think you are wrong to ask questions and to want things confirmed by your GP.
I was in a similar position in November 2011 and had a similar reaction initially.
Ds1 (then 9 years old, built like a strip of wind, with not a scrap of weight on him) developed a nasty cough that wouldn't shift, so I took him to our GP. He listened to his chest, gave him antibiotics and said he thought he should be checked for asthma.
I was surprised and felt there was absolutely no way he could have asthma, but thought he'd take the antibiotics, have a clear check up and that would be that. I was wrong.
A few days later ds suddenly started to really struggle to breathe, it got worse and worse and I called for an emergency appointment. We rushed him in, the GP took one look at him, listened to his chest and rushed him straight down the corridor to be nebulised. He said his chest was so tight he was barely getting any air in at all. I am a bit vague on the details of nebulising but was told it had to be stronger or longer than they expected.
GP prescribed both preventer and inhibitor inhalers and told me ds was asthmatic and would need regular monitoring by the nurses in the asthma clinic.
I was shocked. He had never even had a cough in his life before - except - when he got swine flu during the epidemic and at that time we'd been quite worried about him because he was wheezing and fighting to breathe, but of course being mid-epidemic, the doctors didn't want to see him.
He was referred to the asthma clinic, but by the time the appointment came round a couple of weeks later he seemed fine, although was still using the brown inhaler as instructed.
Knowing nothing about asthma myself, I was suprised when they said he had to continue to use the brown (steroid) inhaler to help his chest and they would review him in the spring. (This was late November-ish.)
I wasn't happy about him having to have daily steroids, but having been scared by how ill he was when he had to be nebulised I didn't question it, as there was no way I would risk him having to go through that again.
Ds was absolutely fine from then until his spring appointment. In fact by spring his peak flow was normal and there was absolutely no indication of asthma or any breathing problems. We were able to wean him off the steroid across the summer, but warned that he should start using it again at the first sign of a cold or cough.
Sure enough, November last year, the same thing happened again. An almost identical cough and a downward spiral from there. We put him straight back on the steroid inhaler and he has had to use his ventolin several times since.
We were told that, given the time of year his asthma flares up, he is most likely allergic to leaf mould and that it's actually a fairly common cause of asthma. The underlying reaction to the leaf mould, plus virus or bacterial infection is a bad combination, creating problems in the lungs which result in the asthma. Provided they don't have any problems the rest of the year, the asthma clinic is happy for them to wean off the steroids over the summer and go back onto them in the autumn or at the first sign of any wheezing or coughing.
Ds does get mild hayfever, mostly affecting his eyes, but doesn't wheeze or cough across the spring and summer. Interestingly he has recently developed eczema on his face and hand as well and last time we were at the GP they mentioned the fact that his allergies seem to be increasing.
I do understand your reaction. It took me quite a while to get my head around ds having asthma and to be honest, if he hadn't had that sudden bad attack and needed nebulising I would probably still have doubts. As it is, seeing him in that state once was enough to scare me for life and I now make sure he takes his brown inhaler religiously.
I think it's a good idea to book an appointment and talk it through with your GP. I know my GP would be happy for me to do this and in fact encourages us to ask questions if we are at all unsure.