The notion of a beat can come across as a vague concept. In mainstream music, when detected, one can tap along to it, in simple rhythm. This rhythm is put into sets, called bars or measures, so that it's a repeating pattern.
The problem comes when, say, in 4/4, there's a strong beat on 1, and a slightly weaker one on 3. 1 2 3 4.Some people would perceive the beat in that music just as 1. Others think 1, 3, while others say it's all four. In fact, in much quicker pieces, 1 will do, and in much slower ones, all four suffice.
But, generally, we tend to think in the latter mode, and this is often reflected in the bpm at the top of a piece. Usually shown with the ubiquitous crotchet as 'the beat'.
You mention 4/4 but with 8 'beats'. That can be shown as 8/8, although 4/4 is far more common, easier to write and easier to read.
Another anomaly is compound time, such as 6/8. Here, there can be found either 6 quicker 'beats' or 2 slower 'beats'. Marching music is found in 6/8, and the left/right is 1 and 4 of the quicker internal 'beats'.
So, a good question that must reflect confusion for many! I guess 'pulse' would be another term used, but not all listeners would agree where that pulse lay in a particular piece, as illustrated previously. Writers need to establish what they think of as a 'beat', in order to write the dots properly, and state a bpm.
It certainly isn't a new beat for every new note played, as you considered, as sometimes, 'beats' aren't even heard. When there are rests, the beat still goes on, so to speak, but nothing can be heard to indicate this, only one's internal rhythm clock, which has, by then, locked onto the tempo and feel, and keeps ticking away until the next sound is heard - which may or may not actually be on another 'beat'. Syncopation rears its head at this point, but I don't want to go there yet...