I don’t think the part of Reina’s phrasing that Xanadu shares is the cause of that limit.
The rez cost of the first piece of ice rezzed each turn is increased by 1.
And the relevant UFAQ says:
Reina’s ability only lasts for the duration in which the Corp is paying to rez the ice.
But I think that limit is just how the “the first piece of ice rezzed each turn” part of Reina’s ability is implemented, and I don’t see any reason why it would apply to cards that increase rez cost more generally.
I think the RNG Key UFAQ ruling about Breaker Bay Grid is more relevant:
If a Runner with RNG Key runs on an R&D with Breaker Bay Grid rezzed in its root, what number does the Runner need to name in order to gain credits from accessing a Vanilla?
The Runner must name -5.
Xanadu’s phrasing is much closer to Breaker Bay Grid’s than to Reina’s, as it doesn’t mention an event that its cost change could be limited to.