GRRM is correct that only a person who is able to enter minds of Wolves or Dogs is called a warg. I do not see how do the examples you cite contradict it.
The Hawk belonged to a "Skinchanger" by the name Orell. Jon Snow kills Orell.
Later Varamyr Sixskins claims the Hawk. And he uses it to scout against the Night's Watch at Battle of Castle Black and before.
Varamyr is the "guard of Mance" that you are mentioning. He is correctly described as a warg because he does have Wolves bound to him. Following Animals were controlled by Varamyr:
- A snow bear
- A shadow Cat
- A Hawk
- Three wolves named One-Eye, Stalker and Sly.
These are the six Skins of Varamyr. And since three of them are wolves, that makes Varamyr a warg, not just a skinchanger.
Depiction of Varamyr with his bear, shadow cat, hawk and wolves by Elena Maria Vacas:
Wolves are special sort of Animals because from ADWD prologue:
Wolves were harder. A man might befriend a wolf, even break a wolf,
but no man could truly tame a wolf. “Wolves and women wed for life,”
Haggon often said. “You take one, that’s a marriage. The wolf is part
of you from that day on, and you’re part of him. Both of you will
If you have any other examples of a Skinchanger without a wolf being called a warg, please share them. I do believe Varamyr is the only example here and you are mistaken about him.
TLDR: As GRRM said, All Wargs are Skin-changers but not all Skin-changers are Wargs.