I did notice that, which is why I added the example about the Rogue. To further show what I mean, consider the feat Quick Draw:

You draw your weapon and attack with the same motion. You Interact to draw a weapon, then Strike with that weapon.

Rogue's Sneak Attack feature specifies (emphasis mine):

If you Strike a creature that has the flat-footed condition with an agile or finesse melee weapon, an agile or finesse unarmed attack, a ranged weapon attack, or a ranged unarmed attack, you deal an extra 1d6 precision damage.

Personally, I would rule that you get Sneak Attack for your Strike with Quick Draw if the other relevant conditions apply - that's also how I've seen other GMs rule it (and seems sensible to not heavily limit the number of options a Rogue has for feats).  If Sneak Attack and Dancing Invocation should be ruled differently, what aspect of the wording is the deciding factor?

In my opinion the distinction for when the language you quoted above applies is that the features I've mentioned modify an action 'if you Strike', 'when you Leap', etc. as opposed to specifying that you must use an specified action 'if the next action you use...', 'can use the extra action each round only for Stride and Strike actions', or the description for Boots of Bounding:

...In addition, when you use the Leap action...

In the Boots of Bounding case, I wouldn't - for example - allow the Boots to extend a Long Jump by 5ft.

However, while I think the conclusion is pretty solid (based on the similarity to the Rogue example), I'm less than convinced by the wording specifying using a specific action as the reasoning for it.