One Speed: When you double move, add the speeds of the two move actions together and then move.
It's not moving your speed twice; it's moving once with a speed twice normal.
A double move has One Speed, which is the speed of the two move actions added together. The speed of the move actions is 2, and the two added together is 4, so the double move has a speed of 4. Taking a double move, therefore, is arguably increasing the character's speed above 2, forbidden by the Slowed condition; his double move has One Speed which cannot, according to the Slowed condition, be greater than 2.
-Hyp.
This can lead, logically, to the conclusion that a double move is no longer possible.
What it does not lead to is the conclusion (which you have stated elsewhere) that the player cannot still take a second, ordinary, move after taking a first move.
Where I think your interpretation goes askew is in taking the non-rules box text: "If you take the same move action twice in a row—two walks, two runs, two shifts, two crawls—you’re taking a double move." as meaning "you cannot take the same move action twice in a row unless you are making a double move."
I think that two consecutive moves can be a double move not must be a double move. Sometimes they are simply two consecutive moves.
That, and it is not clear when you must declare a double move or how that declaration interacts with powers that interrupt that move.
(I also think that the text that specifies that if you have already moved two squares when you are slowed you must stop was written with zero consideration of the double move possibility and thus did not consider that special case. I suspect that rule of being poorly written and that it should have been written to specify "or 4 squares if making a double move." in which case much of this discussion would be moot.)
Carl