Nope, the wiki is wrong. It's Tal-Vashoth.
Varric calls you such in the beginning of the game if you stray from the main road, and so does Josephine in a little interview regarding your background.
People repeatedly mix Tal-Vashoth and Vashoth up in the game. Varric and Josephine aren't exactly immune to this; Varric never encountered any Vashoth in Kirkwall, and Josephine admits to knowing basically nothing of the Qun or the Qunari. Even leading Thedosian anthropological scholars have trouble with the distinction, as highlighted by the Codex entry on the Vashoth; Genitivi repeatedly confuses a woman who was born outside the Qun with "Tal-Vashoth", which gradually irritates her more and more until she cuts the interview off.
Adaar is also explicitly described as "Vashoth" in her Codex in the game, and in her conversations with the Iron Bull she can point this out. Her parents were the ones who left the Qunari, and she was born outside of it: the definition of Vashoth, not Tal-Vashoth. The Bull doesn't really care for the niceties of this distinction and continues to refer to her as "Tal-Vashoth". To a soldier who spent much of his career in a bloody anti-insurgency campaign directed in large part at Tal-Vashoth, protesting that "I'm one of the
nice ones!" understandably doesn't fly. An American GI overhearing somebody speaking in German in Europe in 1944 probably wouldn't really care that the individual was from Switzerland and thus probably
not a Nazi. That does not mean that the distinction doesn't exist.
On a practical level, the writers stated both before and after the game came out that trying to account for all of the distinct varieties of positions with respect to the Qun simply wouldn't be worth it. For much the same reason as Thedosians can't catch the difference between Tal-Vashoth and Vashoth, most players aren't expected to do the same. So they have most characters in dialogue refer to the binary distinction of "Tal-Vashoth" and "Qunari", while making it clear in the Codex that Vashoth are a Thing and that Adaar is in fact one of them.