The confusion may come from everyday's colloquial synonymous use of krank and erkältet. People do know that one is a subgroup of the other but since a cold is such a common cause for being ill we use krank und erkältet synonymously and in the same context:
Ich bin leider [krank geworden / erkältet] und möchte heute lieber nicht kommen.
Ich glaube ich [werde krank / habe mich erkältet].
Bist Du [krank / erkältet], oder warum bist Du so blass?
Usually erkältet is used for a minor disease and reserved for a common cold including headaches, cough or rhinitis. If the cold is more severe, and we may have fever in addition there is a tendency toward using krank but there is considerable overlap.
We would however not use just krank for a serious disorder but would say something like er ist (schwer) erkrankt or er leidet an einer (schweren) Krankheit if we do not want to name the disease we suffer from.