Think of it this way: you are not attacking 3 times. You are attacking once, with 2 copies.
With Multistrike, your 3 attacks would hit 9 times. Ruthless would be applied to the 7th, 8th, and 9th hit.
This is pretty much correct.
When supported by Multistrike Support, Ruthless blows work differently, instead of occurring every third hit, it occurs every third multistrike.
This quoted part is technically incorrect, in that nothing about how Ruthless Blows work is actually different with Multistrike to without. They are simply completely different effects which care about fundamentally different things - and neither of them care about a number of hits.
Ruthless explicitly tracks how many times you use the skill. When you push the button (or leave it held when finishing the previous skill use) and pay the cost of the skill to start doing it, that's a single skill use.
Multistrike causes each use of the skill to repeat twice - each single use of the skill now performs the animation three times instead of once. But that's still one skill use, which is why you only pay the cost once for the whole set of repeats and don't need to consume another cooldown use when it repeats.
Any given use, or individual repeat, of the skill could cause an arbitrary number of hits, which neither support cares about in the least. Multistrike's damage bonus applies to all hits from the specified repeat each time you use it, and Ruthless's damage bonus applies to all hits from the specified use of the skill - including all times that skill use repeats.
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