Spell vamp and lifesteal are not tied to damage type but rather damage source. Abilities such as pan's spear throw will proc spellvamp, not lifesteal.
There are a few abilities that "act" like auto attacks in that they proc on hit effects (gangplank's parrrrley and ezreal's mystic shot) and in this case they will proc lifesteal, not spellvamp, but they are the exception and not the rule.
For the most part onhit effects will never proc spellvamp. however, there are a few abilities that SEEM to act like on hit effects but are counted as spells instead. This includes jax's ultiamte proc, and kassadin's W active (note that being counted as spells, they ALSO proc rylais similarly).
The way items work is kind of odd. But there are few direct damaging items so it doesn't really matter. AFAICT passive effects will not proc spellvamp (thornmail, sunfire) but active abilities will (Death fires grasp, gunblade).