Hiya,
Where to start.
Please, make sure you're comparining the same thing

I don't believe that the Caithness/Italian vases come from the same source as the red vase that started this thread. Yes, they have some similarity, BUT they are not finished in the same way .... note
finished. Yes the battuto technique is basically the same, but the finishing is differnt - and that is critical. Nor does the pattern cover the whole vase, as so often with Caithness when this, and similar, decorative technique is used by them.
I have watched this seller on and off over the months since he started selling his collection of '2000' and have found him a tad wayward on attribution and dating. In this case he should be correct, since the pair of vases have labels, which, sadly he doesn't show. Then he goes on to mention the country of origin is Italy, leaving us with a question mark.
For me there are too many unanswered questions to make this an accurate attribution for the original question. Interestingingly, it does raise other questions about where Caithnnes glass was made, and did they outsource? I have some otherwise Caithness pieces with "Made in Poland" labels on the base which I have been trying to verify..... but that should be for another thread.
Therefore my vote is, NOT Caithness.
Cheers, Nigel