Only physical damage from basic attacks procs lifesteal unless otherwise stated. If an AA has on hit true damage or magic damage, this does not proc lifesteal.
All types of damage can proc spellvamp, so long as the damage comes from an ability that procs spell effects. Jax w and R passive proc spellvamp because those abilities specifically proc spell effects. Jax is the only champion i know of that can proc spellvamp from basic attacks.
EDIT: okay there is a handful more champions but they arent really viable because most of them apply it in AOE fashion or it is some long cooldown. Jax has consistent permanent single target spell autos