You might be overthinking this
Clarifying, it's really just a good cleaning that removes build-up. That's it. You can clarify at home by adding some baking soda to your regular shampoo, it's that simple. Or you can get a shampoo that has the word "clarifying" on it.
With that being said, if your salon tells you they have a clarifying shampoo, it's safe to say they have a clarifying shampoo. If you suspect that they're not being entirely truthful about it (although I can't imagine a reason why they'd lie about something like that), you can ask to see the bottle.
The product you just linked to, I really think "detox" is their fancy pants, trying-to-squeeze-more-money-out-of-you-by-clever-word-usage way of saying "clarifying. The thingy says it has baking soda in it, that's good enough. I didn't look at the rest of the ingredients, but it looks fine.
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