I also go under because it cinches the string sooner and makes it easier to ensure the tails slant downward toward the top.
I like to make all my tails point inward (and down into the fretboard). After you thread the string through the bridge hole and pull it back up to the bridge channel, pass the working end behind the standing end (the rest of the string) in the direction opposite to the one you want the tail to eventually point, then start the windings by passing the working end through the "loop" you just formed—"going under". If, before you tighten, you hold the tail in place with your thumb on the underside of the bridge (where the holes emerge) at just the length you eventually want, you can ensure the tail points as desired and no excess length has to be clipped—an important consideration if you fuse or ball the string ends. What you want is for the string to cross over the bridge edge before it passes under the "upward-going" part of the string for the last time, so that it's cinched on the underside of the bridge, right by the bridge hole, not on the wider flat where all the other winds lie. For wound strings, this final passing is often the only passing, and cinching on the underside is all the more important. Just "pulling tight" is often a bad approach because it's more likely to leave the final cinch on the top of the fretboard.
For what it's worth, concerning the bead approach, knot experts say that a double overhand knot and a figure of eight knot can both pass through the same size hole that a single overhand knot can; better to use an Ashley stopper knot—or, if your knots are holding, then a single overhand knot should probably suffice. I think the single knot looks better than the bulkier knots (especially when using small beads), though with a double overhand knot you may be able to clip off the excess closer to the knot, minimizing any arm- or sleeve-snagging incidents.
The number of winds also depends on the string thickness and the width of the flat part of the bridge in the direction that the strings run, which varies from bridge to bridge. Too few winds and the string can pull loose; too many and the final twist will buckle out, causing the string end to point in the wrong direction.
Apologies for the thread creep.
