The more I think about it, the more convinced I become that the slur is the critical marking in music, but that different kinds of slurs mean different things.
For example, a slur over two notes indicates a slide from the first to the second, usually down, but not always. The note value of the sound produced is the sum of the two slurred notes. So a half-note slurred to a quarter becomes a dotted half-note, or a half-note that slides onto the quarter-note pitch. Also, it is important to note that the release of that quarter note will not coincide precisely with the beginning of the following beat, as it is weak rather than emphatic.
The three-note slur may be an indication of an ornamental slide familiar from the Baroque period when a curved line before a note indicated such a lead-in to it. These are not actually rises to the top note, though that note may be notated with a longer value than the two leading up to it. Rather, it is a weak pulling away from the lowest tone, which must remain the most "real."
More problematic are the slurs over several notes, of which I can cite two problematic examples:
In the case of four or more notes of like value I check whether the last note is higher, lower, or the same as the first and feel that overall direction as the point of the slur. Hardest by far is when the notes match and I must "move" without going anywhere. (Schumann: Einsame Blume)
When there four or more notes of unlike value the problem is more difficult.. I have in mind the opening of Beethoven's "Spring" Sonata for Piano and Violin, which opens with a half-note slurred to eight sixteenths that seem to fall. But how can a line convincingly fall from such a sustained value through so many quick values? I maintain that it does not fall, but rather that it is wisped away, so to speak, at the end of the upbow, permitting the true counterpoint of the melody to assert itself. In the sense of "like goes to like," then, the half-note A goes via the wisp to the half-note C and so on. It makes sense when you think about it but only if you imagine playing the entire bar without counting to four.
