Jan
In music many things are about convention rather than absolute logic. A slash chord is just a convenient shorthand way way of telling the player to play the chord with something other than the chord root as the bass. Is it an extra note or a replacement note? - it really doesn’t matter. Of course that new chord could also then be termed an inversion. But there is no convenient way on a lead sheet of transmitting to a player ‘play a second inversion’ other than writing it as a slash chord like C/G
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Well, not to throw even more confusion into the mix, but I have actually occasionally seen a form of notation that indicates the inversion of a chord, and that is to have two numbers, one on top of the other, right after the chord symbol, that shows the intervals that make up the chord, the top number being the "span" of the chord, or interval from the bottom note to the top one, and the bottom number being the interval from the bottom note to the middle note being played.
So, for a C major chord, a C 5/3 (I'm just adding a slash because the text format here doesn't easily allow for showing one number on top of another, but in the actual notation, the two numbers would be one over the other, and there would be no slash) is root position, as it shows the "bottom to top span" being a fifth (C to G) and the middle note (E) being a major third above the bottom note. C 6/3 is first inversion (the "span" of the chord being the E-to-C minor 6th, and the middle note (G) being a minor third above the bottom note. And C 6/4 is second inversion (the sixth being the G to E span, with the middle note (C) being a fourth above the bottom G.
Confused yet :-) ?
But with the original question about the slash chords -- Yes, no need to overthink it. The symbol on the left of the slash is simply the chord being played, as it would be played any other time with or without the slash, and the note after the slash just indicates what bass note is to be played along with that chord. And it is typically used when the song writer wants to indicate that it is important to the sound and character of the song that a particular bass note is played along with a particular chord at that moment in the song. And, as previously stated, the bass note does NOT have to be a note that would normally be a part of the chord -- and in fact, it often isn't, which is why the song writer wants to make it clear to play that particular bass note with that particular chord.