Yes technically true. HMCG are recognised in international law as responsible for maritime search and rescue operations in UK territorial waters. Practically speaking in most cases if you have a problem in UK (and Irish) waters the RNLI is probably going to search for and rescue you, if it’s really bad you’ll get a helicopter dispatched as well from Bristow helicopters who are subcontracted to HMCG but I think there’s only about 6 choppers covering the whole of the UK.
HM Coastguard and the RNLI do different roles though - the former is more about co-ordination amongst different rescue agencies and the RNLI runs lifeboats.
But i still maintain HMCG best fits the description "the maritime search and rescue service for the UK". I mean, if you call RNLI with an emergency they'll tell you to hang up and dial 999.