I think new Warlord's should take all on-hit damage into consideration.

So what new Warlord's basically does is that movement and basic attacks give you charges, after about 6 auto attacks while moving you will be fully charged and your next basic attack grants up to 50% lifesteal and 30% movement speed for 0.75 seconds.

I really like the concept of that. Currently, Warlord's is much less interactive than all the other keystones to the point where you don't even have to pay attention to it, basically just another statcheck. New Warlord's makes you think about on what you actually want to spend that energized 50% lifesteal auto attack on.

For example, Ezreal wants too keep that in mind to ideally let Warlord's proc on his Q rather than just a basic attack since it deals more damage and therefore heals you more. Same with Yasuo's Q, Draven's Q, Caitlyn's Headshot, Vayne's Q, Nasus' Q etc. which all proc lifesteal and deal more damage than your usual auto attack.

And since they all get their special auto attack to proc Warlord's I find it somewhat unfair that others don't. For example something like the bonus damage from Aatrox Blood Price W, Jax' W and ult, Yi's E, Kog'Maw's W any many more don't proc it even though they have about the same power level as the other abilities mentioned above.

Ultimately this change wouldn't really make it stronger or weaker, it would just let Warlord's Bloodlust be accessable to a lot more champions and just add to diversity and the meaningful choices you are supposed to make with your masteries.