Having been through several rounds of comments with the OP (and earlier versions of this answer, for which many thanks, quadratecode) I've come to the position that the EUPL does not apply to SaaS in the way that the AGPL does. However, it's also clear that the licence authors think it does; they have so written in guidelines and other prefatory material.
The problem is that they haven't written anything in the licence text itself that clearly and unambiguously achieves that aim, in the way AGPLv3 s13 does; instead, they've tried to interpret (and to a lesser extent, redefine) "distribution or communication" in such a way that it covers SaaS, which no other free licence does. Furthermore, they use some very odd language to achieve that end ("Distribution ... is ... making available, online or offline, copies of the Work or providing access to its essential functionalities"), language which appears in no other free software licence I know, and which is itself undefined in the EUPL. New language definitions, which require courts to lay down new interpretations, are not as good an idea as using simple language that clearly covers the desired activity ("your modified version must prominently offer all users interacting with it remotely ... an opportunity to receive the Corresponding Source").
Courts have a strong tendency to read licences on their faces, that is to say, they look only to the text of the licence to determine rights and obligations. In some cases, courts have explicitly refused amicus briefs from the authors of the licences who wished to clarify the intended interpretation of their text. In short, if it's not in the licence, you can't rely on it, no matter what the licence author says about their intent.
By the OP's own admission, the issue of SaaS applicability is very unclear until the prefatory and guideline material is consulted, and on that basis, I would not expect a court to leap to the conclusion that SaaS is a covered activity. I'm not saying it wouldn't, but it seems a lot less reliable a conclusion than in the case of AGPL.