"Mayday call. Vessel in immediate distress. " That's not the way I've seen it defined by the FCC or USCG. (Eldridge actually says if "...You are in distress...threatened by grave and imminent danger" you, not the vessel). Every formal definition I've seen of MAYDAY defines it rather simply as the EMERGENCY call, and further state and federal laws further define "emergency" as a situation where there is "imminent risk of loss of life or property". I mention especially the FCC, because they regulate all radio transmission in the US, and by both FCC regulation and international convention, "Mayday" is specifically defined and used that way in all radio services--not just the maritime ones. A Mayday call invokes and receives special rights on any radio frequencies.
What Chapman or Eldridge or anyone else define "Mayday" as, doesn't define it until they relieve the FCC of duty and take over the watch.
SECURITY and PAN are not emergency calls, that's the difference. Eldridge calls "PAN" an "Urgency" call, which might be a good way to describe it--expectable danger is coming, just not imminently.
The bastardization "Mayday" derives from the French term "M'aidez" meaning, literally, "aid me" or "help me".
That includes situations where the vessel is not at risk, but someone onboard is, i.e., a steam pipe bursts and gives crew 3rd degree burns requiring medevac. Or, your trimmer has just amputated his fingers in a genoa winch. He'll live, the boat is in no danger, but an immediate medevac might re-attach those fingers. Or, you've witnessed a collision between two other vessels, and they are on fire or sinking with crew in the water.
Those are all emergencies, all Mayday situations, none of them involve distress to your vessel. You're still entitled and encouraged to call Mayday in all of them.