Don't worry - I'm pretty sure vinegar, even malt vinegar, is pretty low in carbohydrates.
A quarter cup has something like 3 g carbs, and that's a lot of vinegar.
Regular barley, like a big bowl of barley cereal, should be avoided - that's correct. But malt vinegar is basically highly fermented malted barley, so you should be fine. "Fermented" basically means bacteria have transformed the sugar and starch into carbon dioxide, alcohol, acetic acid, etc. For example, fermented milk (high in carbs) becomes much-lower-carb yogurt. Fermented barley usually becomes beer.
Barley fermented with aceto bacteria becomes vinegar. (I'm a homebrewer, I know these things.
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The nuts, I wouldn't worry about those, either - just try to avoid them until you're off induction.
Other plans like Protein Power permit nuts right away - the cool thing about induction is that since it's more strict, you get "breathing room" that allows for a mistake or five, and you'll lose the weight just fine.
Enjoy!