Ah now. Beer is something i do know a little about.
I think the word "Beer" is the general term for all types.
You can then sub divide that into lagers, dark beers, wheat beers, stouts, milds, pale ales, bitters, golden ale, etc etc
But in general we normally divide Beer into two main classes; Ale and Lager.
Most beers until relatively recent times were what are now called Ales. Lagers were discovered by accident in the 16th century after Beer was stored in cool caverns for long periods.
Most "Beer" that you drink in the USA is in fact Lager.
You can also sub divide Ale into Keg Ale and Cask Ale.
Cask ale or "cask-conditioned beer" is the term for unfiltered and unpasteurised beer which is conditioned and served from a cask, usually without additional nitrogen or carbon dioxide pressure. Cask ale may also be referred to as
real ale, a term coined by the Campaign for Real Ale (CAMRA), often now extended to cover bottle-conditioned beer as well.
Keg Ales are pasteurised (which makes them last longer) and are stored in CO2 pressurised Kegs.
When I started drinking beer in the 70s it was very hard to find "real ale" anywhere. Almost all beer was keg and quite frankly it tasted crap. We just didn't realise it was crap. But CAMRA slowly put pressure on the big breweries to let some of their pubs sell low volume ales made in small independent breweries. As these caught on the big players bought up these small fish and started making the stuff themselves.
These days all the beer is of quite good quality. Real cask ales are easy to find in almost every pub now. Some pubs even make their own beer on the premises like in the olden days.
To answer your last question Eddie. All beer is cooled. Or at least it should be. But different types of beer like to be cooled to different temps. Lager that you guys drink over there likes to be VERY cold. Ale however starts to loose its taste if it gets too cold so although not "warm" it is not served as cold as lager.
Also. In the days before refrigeration the casks were only cooled by having wet sacks placed on top of them. This was still going on in some places during WW2 and is probably where the myth in the USA comes from as GIs would have gone home with these tales.