So do you have to have that border for the avatar to count?
Technically, no. There is nothing in the intro text that specifies that you must use exactly the same border as every other avatar in the Org's collection.

That said, the idea of the contest is to make avatars which will be added to the Org's collection, with Org mods as the judge of what fits. It would be wise to use the same border, or at least one that is quite similar in saturation and pattern.