I'm looking into what Planeswalkers abilities are considered. Are they considered activated abilities or loyalty? For example I know that Pithing Needle works to stop Planeswalkers from doing anything. With that logic does Illusionist's Bracers also cause Planeswalkers abilities to be copied?
Yes. Planeswalker abilities are both activated abilities and loyalty abilities (loyalty abilities are a subset of activated abilities).
Bracers can indeed copy an activated ability of a planeswalker assuming it is a creature and equipped (as is Gideon in your scenario). So, yes, you can make him a creature, equip him, and activate his -4 to end up getting 2 Emblems.
Loyalty abilities are activated abilities, yes. That's why Pithing Needle can stop them, and so for the same reason you can use Illusionist's Bracer's to copy them, IF the planeswalker is also a creature and equipped with it. In your example you will indeed get two emblems, one from the copy and one from the original loyalty ability.
For example.
I have a Gideon, Ally of Zendikar and a Oath of Teferi out. I activate Gideon to become a creature. Then I equip Illusionist's Bracers to him. Then activate his -4 ability. Do I create 2 emblems?
With how the current wording is I'm assuming yes.
Bracers can indeed copy an activated ability of a planeswalker assuming it is a creature and equipped (as is Gideon in your scenario). So, yes, you can make him a creature, equip him, and activate his -4 to end up getting 2 Emblems.
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