Your conclusion that this is not asthma, because you have no expiratory wheeze, is probably correct, especially since you are keenly aware of the symptoms being inspiratory. Inspiratory noise, whatever term one uses to describe it, is usually an indication of what we call upper airway obstruction; the obstruction located in the mouth, throat, voice box including the vocal cords or the windpipe. The obstruction can be caused by a mass, by large airway collapse, by vascular disorders or by a functional disorder such as spontaneous vocal cord paralysis or a condition called vocal cord dysfunction (VCD).
You should arrange for consultation with a specialist, either a pulmonologist or an ENT doctor, for a direct examination of your voice box. Such a specialist might want to conduct the examination before and after exercise and may want to get a magnetic resonance image (MRI) of your upper airways, including the voice box. A good, non-invasive screening test for upper airway obstruction is a simple breathing test called spirometry with both inspiratory and expiratory flow-volume loops.
You should discuss the above with the consultant and not put-off the evaluation very long. We would welcome a follow-up from you.