When is an attack the Attack action?
Distant Strike is pretty clear - it says Attack action with the capital A. It is not any action that allows the character to make an attack (such as Green Flame Blade, which includes an attack as part of the Cast A Spell action).
At 11th level, you gain the ability to pass between the planes in the blink of an eye. When you take the Attack action, you can teleport up to 10 feet before each attack to an unoccupied space you can see.
The Attack action and the ability to teleport are part of the same sentence - "when" and "each" ties the two together. If you're not currently in the midst of the Attack action, you don't benefit from Distant Strike.
There are no hidden rules. If the ability were meant to work in conjunction with anything but the Attack action, it would say so. Perhaps something like "On a turn you take the Attack action..." or "each attack during your turn", but it doesn't use more permissive wording.
Take Reckless Attack for example:
When you make your first attack on your turn, you can decide to attack recklessly. Doing so gives you advantage on melee weapon attack rolls using Strength during this turn, but attack rolls against you have advantage until your next turn.
It has to be activated before the first attack and it specifically affects all melee weapon attacks: it doesn't require the Attack action. A Dual Wielding Fighter Barbarian could combine it with Action Surge, and all the attacks would benefit. Opportunity Attacks, however, would not - because it ends when the turn ends (the downside lasts longer).
That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
Haste allows the Attack action (capital A), but has a special caveat. It is still the Attack action, so it meets the requirements of Distant Strike.
Bonus action attacks? No.
I am unaware of any bonus action that allows a character to take the Attack action (again, capital A), just ones that allow the character to make an attack (lower-case).
If such a thing did exist, it would potentially allow for a Fighter to potentially attack twelve times in a round - regular Attack action and this hypothetical bonus "Attack" action for four each, plus Action Surge for another four.