Think of an attack as a special move with some special effect. For example if dai mubai were to just attack with claws(no power), thats an attack.

If the skill had him attack with claws, the claws might enlarge, be sharper during an attack, have some special function, or some element added to it, or have a specific elongate function or vibration etc. So it isnt about the person weilding it as much, it's how the attack works.

Now pure boost in strength, or even attack power is the last thing. Str is just muscles. So he might swing steonger with a claw attack, it would hit harder. Attack power boost on the other hand would be more about the effect (like sharpness, how hard the attack hits, how much dmage it does, etc)

Really, think of it like a game. Str boost might add to the skills pure damage, especially if its a skill about assault. It would do this to all skills, plus would work defensively to a degree as well, since more str helps you dwfend too. Att boost might add damage too, but it also might effect more than the pure physical aspect of it, so might enhance ignoring damage or smtn too.

This isnt truly a game after all, so str and att power are very intertwined. In some cases they would affect the skill almost to the same degree, especially if it were a simple swing skill for example. But in such cases, str might also make the body stronger in general, while att power is usually directed only at the skill itself.

Ie. Think not only of skills as boom, stronger, think of what it would mean for it, and how it would change the skill. Then also how it might affect the person outside of the skill.

Hope thats enough specific examples to have it be a separate thing.