The simple answer is no, there are no "subclassifications" of emergency or urgency calls in standard aviation phraseology. In terms of priority, they are all the same; however, in terms of actions taken, there can be differences based on what caused the aircraft in question to make the emergency or urgency call.
There's mayday (emergency), pan-pan (urgency), and everything else (routine traffic). Mayday trumps pan-pan, and pan-pan trumps everything else.
- mayday is used when the safety of the flight has been compromised and the pilot(s) require immediate assistance to recover.
- pan-pan is used when the pilot(s) require assistance beyond that normally offered by air traffic control, or require priority handling, but the safety of the flight itself has not been compromised.
Compare When should the term “pan-pan” be used instead of “Mayday”? which discusses the difference between the two in more depth.
Yes, there is a bit of overlap between the two: it's possible to come up with situations where it is not entirely clear whether "mayday" or "pan-pan" is more appropriate. When in doubt, it becomes a judgement call. A pan-pan can escalate into a mayday as the situation develops, and even a mayday can be terminated if the situation that called for the original mayday call no longer exists. (The latter could, for example, happen if a GA VFR pilot becomes uncertain of their position, and are then able to identify their position in such a way that the flight can proceed safely; or if a non-instrument-rated pilot flies into IMC, and are able to return to VMC and proceed safely.)
The broad classification of emergency or urgency is then further specified by describing the type of emergency or urgency situation, using either standard phraseology or plain language, as appropriate. The purpose of this, however, isn't to prioritize such calls, but to identify to ATC (and other aircraft) what types of assistance is required.
Compare How does ATC handle different kinds of emergencies?, though that question is somewhat US-specific and specific details can vary between countries.
To mention a few examples, without in any way intending for this to be an exhaustive list:
- mayday, VFR into IMC, negative instrument qualified — this one likely requires immediate help in returning to VMC, whatever that entails
- mayday, decompression, emergency descent to ten-thousand — no immediate assistance required for that flight, but get everyone else out of their way now, then figure out how and where to get them onto the ground
- pan-pan, fuel — this one needs landing priority, and may require vectors to the nearest suitable airport, but not much more than that
- pan-pan, passenger ill — this most likely requires notifying someone on the ground so that the situation can be handled when they land, and might require this one flight to divert, but other than that, not a big problem from an ATC standpoint