Actually.... If B requires A does not necessarily mean A causes B. If I am required to have my car keys on me to drive my car, having my car keys on me does not mean that I am driving my car.
Except that's not what we're talking about. The statement is pretty clear.
'When you take two of the same move action in a row, you are taking a double move.'
Your counter argument is appropriate for the form B -> A, but the statement in question is of the form A <-> B. The rules say, firstly, when you take the same move action twice in a row, you are taking a double move (A -> B) then to take a double move, you must take the same move action twice in a row (B -> A); (A -> B) and (B -> A) becomes (A <-> B).
I can have my keys and be eating dinner at home or riding a bike in the park. A double move requires you take two of the same move actions in a turn. But nowhere does it say that taking two of the same move actions in a turn is automatically a double move.
It says that taking two of the same move action in a row IS a double move. What is a double move? It's when you take two of the same move action in a row. Are you taking two of the same move action in a row? Then it's a double move.
This is mutual implication, rather than unilateral implication.
If you choose to not double move, you can't end one action in an ally's square, you can't normally shift into difficult terrain, etc. I could still Shift - Shift across clear terrain with two separate actions.
If you take the same shift action in a row, you're taking a double move.
Take a step back and look at a more concrete example. Lets say I have three actions in a turn: One that takes no real time (minor), and two that take 3 seconds each (move and standard). I draw my dagger (minor). If I then take one of my 3 second actions and hustle, moving 6 squares, I make lots of noise and can't easily make the required effort to stealthily move. I then take 3 seconds to move 2 squares, slowing myself down quite a bit, moving quietly, taking as much cover as I possibly can. Doesn't it make sense that way? [I draw my assassin's blade, jog across the alley and slip behind the boxes where my target can't see me.]
What makes sense, and what is RAW are not the same thing. I've argued your way can make sense in some games, and DMs should do it that way. I've also argued that RAW gives a strange corner case, and it creates an odd interaction. But, RAW, regardless of how you want to present it... two of the same move action in a row = a double move. You seem to think that it is optional.... it is actually not.
"On your turn, you can move twice if you take a move action instead of a standard action. I
f you take the same move action twice in a row—two walks, two runs, two shifts, two crawls—you’re taking a double move."
In your example, you're taking the walk action, then taking a walk action. That's walk actions twice in a row. By the rules, you're taking a double move.