Merseyrail's rail only day tickets have something like NORTHWIRRALLINES written on them, rather than Merseyrail Only. The Merseytravel day tickets, the Saveaways being spoken about, are valid on the "city lines" anyway. So I don't think it particularly matters about the supposed confusion. It's never confused me any.
A single to NLW is £4.80 from LIV, so if two journeys are required from North Liverpool to Newton-Le-Willows, the sensible choice would seem to me to be a £4.80 peak single to cover that leg, plus the Saveaway ticket to cover the other three journeys. May be annoying for it not to cover that other peak journey, but it is in any case still a great value ticket easily comparable to the Travelcard in terms of scope, range and flexibility.
I see also someone did point out an alternative of a walk, a bus and an earlier train from LIV. As Saveaways are valid on the buses, all the journeys could be done for the fiver or so it costs for that Saveaway.