Basically, you're right with all your points.
The level between different albums can indeed be dramatically different, depending on how agressively it was mastered. Records from the early days of CDs are often very quiet (engineers celebrated not having to worry about vinyl's noise floor anymore, and also, early CD players didn't really work reliable with full-scale peaks). Then, until recently, there was a steady trend to master ever louder, with not so good consequences to quality. PCM has a hard restriction on what levels are permissible. If you just increase the volume, you will indeed run into digital clipping which sounds completely horrible, but there are tricks which avoid this; basically all about compression. Brickwall limiters are able to to get virtually all the sound to just-barely-clipping level: enough to make the meters go red, but not really audible as distortion.
Fortunately, the record companies have by now realised that this is not a good idea. Not only is it doubtful whether loud really sells better, also, nowadays a louder recording will in fact not necessarily play back louder – because many audio players do in fact “correct the levels or something”, it's called loudness or RMS normalisation.
Also note that, even without normalisation, different-looking levels may not actually be so different: loudness is a logarithmic scale. 6 dB is not a very dramatic difference in loudness, but correspends to doubling the peak amplitude.