Yes - exercise induced asthma sounds spot on. Probably triggered by your recent upper respiratory tract infection leaving your small airways more sensitive. You may find that this goes away completely in a few weeks, or it may be something that stays. A couple of puffs on the Salbutamol inhaler 15 minutes before exercise will help. Asthma is characterised by difficulty exhaling, which results in your lungs becoming overfull and then you can't breath in. People get scared by not being able to get their breathe - but you need to concentrate on getting the air out of your lungs fully so you can get fresh air in. Cold weather makes exercise induced asthma worse, so gentle warm up exercises can help. In really cold weather, breathing through a scarf or similar keeps the air warm and moist so can reduce symptoms. If your problems become more than occasional, a steroid inhaler is likely to be helpful.
A upper respiratory tract infection can give exactly the same symptoms as the OP describes. I am not a doctor, however I was on medication for asthma for more than 30 years and off it for more than 10. I've had more than my share of bacterial and viral chest infections, including one last week.
Salbutamol is a bronchodilator and should be taken in the event of symptoms of an attack. Becotide was the preventative of choice in my day. This was taken regularly to avoid having to use salbutamol, which can affect the immune system to the point where the body is unable to regulate its own natural immune responses and a physiological dependency can occur. So it really is something to take as little as possibly, whilst taking enough to control symptoms of attacks.
This drug works through the process of bronchodilation because asthma symptoms include bronchoconstriction, which is a tightening of the airways. The result is that oxygen cannot be taken into the body and the hypothalamus, which controls autonomic systems within the body including maintenance of a well balanced oxygen to CO2 ration, triggers the 'I can't breathe' response which can quickly escalate to panic, and an attempt to forcibly reventilate which is difficult given the bronchoconstriction. This leads to distress, rapid shallow breathing, and in extreme situations can make it difficult to self administer salbutamol. So again there is the danger of taking too much of it when merely a bit out of breath, and the danger of not taking it quickly enough when a real attack is in progress.
One approach to controlling asthma is to learn to deal with the 'I can't breathe' feeling. If the panic and distress are severe, then the resultant asthma attack can be dangerous, especially if salbutamol cannot be administered. If the 'I can't breathe' feeling can be experienced with a state of mind of 'I can't breathe right now, but I should remain calm and relaxed and go with it', much of the panic and distress can be avoided which can make administration of drugs easier, or even allow the attack to pass without treatment.
*This is a very simplistic statement to illustrate a process and it is accepted there are many different forms and severities of asthma where this process may be very different*
This approach is called the Buteyko Method. And E&OE, I'm happy to be corrected on any of the above. It's a long time since I did in depth research on this. But your point about concentrating on breathing out alarms me greatly. Unless you can clarify in some way, it flies in the face of everything I've ever read or experienced as a 'lifelong' asthmatic.