Yes, the deal matters.
Dumbledore made the deal with Snape to destroy the wand's power. If all had gone as planned, Dumbledore would have been the wand's last owner. (Drawing from The Boy Who Lived's excellent quote):
"Aren't you listening? Snape never beat Dumbledore! Dumbledore's death was planned between them! Dumbledore intended to die undefeated, the wand's last true master! If all had gone as planned, the wand's power would have died with him, because it had never been won from him!"
Harry Potter and the Deathly Hallows
But Draco (unexpectedly) became the wand's owner.
"The true master of the Elder Wand was Draco Malfoy."
Harry Potter and the Deathly Hallows
The wand changes ownership when the previous owner is defeated. This includes disarming, stupefying, or killing. This can happen accidentally, as seen with Malfoy who was clueless to the fact that he was the owner of the Elder Wand until he heard Harry tell Voldemort.
Killing
"That very night, another wizard crept upon the older brother as he lay wine-sodden, upon his bed. The thief took the wand and, for good measure, slit the oldest brother’s throat."
Disarming
“You still don’t get it, Riddle, do you? Possessing the wand isn’t enough! Holding it, using it, doesn’t make it really yours. Didn’t you listen to Ollivander? The wand chooses the wizard. . . . The Elder Wand recognized a new master before Dumbledore died, someone who never even laid a hand on it. The new master removed the wand from Dumbledore against his will, never realizing exactly what he had done, or that the world’s most dangerous wand had given him its allegiance. . . ."
Stupefying
And now Harry was hurrying along a dark corridor in stout little Gregorovitch’s wake as he held a lantern aloft: Gregorovitch burst into the room at the end of the passage and his lantern illuminated what looked like a workshop; wood shavings and gold gleamed in the swinging pool of light, and there on the window ledge sat perched, like a giant bird, a young man with golden hair. In the split second that the lantern’s light illuminated him, Harry saw the delight upon his handsome face, then the intruder shot a Stunning Spell from his wand and jumped neatly backward out of the window with a crow of laughter.
Harry Potter and the Deathly Hallows
What would happen after?
Dumbledore would no longer be the master of the Elder Wand. He could still be able to use it, like Voldemort, but the Wand won't let him utilize the full power [see above quote for "Stupefying"]. If anyone were to disarm Malfoy, the Wand's master would change. But if anyone disarmed Dumbledore, the Wand wouldn't do anything.