In modern parlance I (BE) don't think there's a difference between them. There are regional and historical differences, obviously. I'd be interested to hear what others say. This is how I would categorise them:
living room -- common term
lounge -- common term
sitting room -- more old fashioned
reception room -- the generic word used in adverts for houses for sale (and usually also includes the dining room)
drawing room / parlour -- archaic
front room -- the word my grandparents in the North East used for their living room that was only used for 'best' (even in small two-bed terraces). We did all our 'living' in the dining room. I only mention 'front room' because it's in the dictionary examples above. I imagine it's still in use now in the North East and elsewhere.
Estate agents normally just give a number of 'reception rooms' which would include the room you'd expect to use as the lounge, and the dining room. So the mansion in your example would just have five reception rooms. Here are two examples plucked out of thin air for pricey UK houses: https://www.rightmove.co.uk/properties/84542365#/, https://www.rightmove.co.uk/properties/84245044#/. Notice that the second property in Bath is listed as having 'four reception rooms' but if you look at the floorplan (wow!) it has a dining room, a drawing room, a withdrawing room and a snooker room. But that's borrowing from the language of era the house was built (Georgian). This one here is a more modern house: https://www.rightmove.co.uk/properties/99402677#/. The summary and description just talk about 'reception rooms' and 'living area', but they've used different terminology on the floorplan (sitting room, living room, snug, dining room, lounge).