Usually, in the context of these privacy-preserving payment systems, online vs offline refers to whether the merchant has to be online to check if the 'coin' they received is valid (authentic and not doubly spent). The user usually has no reason to be online at all, since they withdrew the coin already in the past.

By that definition neither the wallet holder nor the merchant would have to be online for a real 'offline' system.

GNU Taler e.g. is an online system on the other hand, where the merchant has to be online for pragmatic reasons. It's kind of sad to see them being categorically excluded by this requirement. Their the best we currently have afaik.

(Check out my answer below for sources https://news.ycombinator.com/item?id=36520725)

That could very well be what they mean, though that isn't offline at all and is no different than my offline credit card being run through an online card reader

If they want to compare it to cash, I'd have to be able to give you the money directly without any network verification. Just the step of a receiver having to ask the network to validate means there is no guarantee of privacy.