This one has WiredSport written all over it.
Anyhow, here's my take until he shows up. Waist width is probably one of the least useful measurements typically provided by manufacturers. I believe it is primarily intended to be an indication of what foot size the board will accommodate, but it is far from optimal for that since it is measured at the middle of the board, not where you stand. It seems it would be far more logical to provide width at each center set of binding mounts, but tradition remains. Things like stance width, sidecut, binding and boot dimensions, even riding style, all affect foot size/board compatibility. Waist width itself tells you none of the things you mentioned. The only riding characteristics even suggested by waist width are what Rookie and Bparmz stated, narrower is generally easier turning/quicker edge-to-edge, and wider floats better in pow. Even that generalization is based on all other factors being equal (i.e. comparing a wide vs regular version of the same board).
EDIT: Oops, I just re-read OP's post and realized I missed the title. Factoring in tip/tail width does tell a little bit more and can affect turning/carving/straightlining, but I'm too lazy now to jump into that one so I'll defer to others.