Others have explained the broad strokes of the mechanisms at play here, so I'll address the one noted peculiarity: growth on the connection rod / electrode.
This may simply be that, being longer, it gets hotter towards the end, and thus participates in the halogen cycle.
Leads are usually molybdenum, AFAIK, so are reasonably good (electrical and thermal) conductors, but a long one will have a higher temperature rise at the end. It probably has a higher temp rise along its length as well, due in part to radiation from the adjacent filament, and perhaps convection through the tenuous gas as well.
This seems at least partly supported by the amount of growth varying with position, being less at the top (cold end towards glass base); there also seems to be a line-of-sight element to it, hence the greater deposition adjacent the filament. (Or perhaps it's not actually much more material, but the deposits happen to be of a taller, flaky nature here?) The pressure inside these bulbs is not exactly a hard vacuum, but it may still be low enough to see a concentration gradient even over this short distance.
And, the other electrode does indeed have a very small amount of growth on it. So it's at least safe to say, it's not purely an electrolytic sort of reaction.
And, that's about the limit of my knowledge on this topic. I don't know offhand if thermionic emission or ionic conductivity plays any role in the halogen cycle. There are certainly atoms and molecules present which could be ionized (whether by electrical discharge, or thermal dissociation). Even if present, it does not appear to be the dominant material transport mechanism, at least.
Sodium for example is readily ionized even at mild flame temperatures -- hence the ubiquitous orange sodium glow when placing things into a fire (there's inevitably traces of sodium on things, for various reasons: sweat, dust, water, etc.).
But these are not sodium lamps, and I don't know what the ionization potentials or dissociation constants are for the species involved here. They may instead be rather stable molecules, which only break down autocatalytically on hot metal surfaces (and at rather high temperatures, at that!). If you have additional curiosity in this direction, perhaps the Chemistry Stack would have some thoughts on it.