I'm sure if the approaches could be made safely they would have, a little to a lot of rain from an otherwise benign system is a normal challenge. If however as you say there were thunderstorms in the area there is a distict possibility that the outcome of the approach might be in question. Thunderstorms are not to be taken lightly. They spawn tornadoes, wind shear, microbursts, unexpected crosswinds, wind shifts, hail, contaminated runways, etc. Most airline SOP's that I have come across forbid landing if a thunderstorm is occuring at the airport.
As to your second question from a pilot perspective there are no degrees of an emergency as far as declaring one. You declare an emergency to get priority handling, be it an engine fire or a ill pax. You of course advise ATC as to its nature. What they do as far as sub categorizing between medical and otherwise and their response to each will have to be answered by a controller.