The term "Asthma" seems to bounced around quite liberally, sometimes in the abscence of a firm diagnosis.
A kid with a single episode of viral wheeze sometimes gets labelled as being asthmatic, with a resultant negative effect on their future career.
We'll always be willing to help straighten this situation out, but Kiev, I'd advise against concealling an episode which required a hospital admission.
Sounds like the authorities are trying their best to get you through. I'd imagine that they'd be interested in how your asthma has affected you, (control, affect on activities etc), since 2004.