variable current rating vs fixed current rating from dc battery, does both of them exist or only one
The mAh rating of a battery is not directly related to the maximum amount of current it can supply. It specifies capacity, not current.

The problem with your application is that typical 9V block batteries can only supply very little current (maybe like a few hundred mA - notice difference between mA and mAh!).

The maximum current is largely determined by battery type. Different battery types have different internal resistance, which limit the maximum current. Lithium batteries for example can generally deliver a much higher current than standard alkaline batteries. But batteries usually don't tell you their maximum discharge current like power bricks do.