Classically it was branded as the Merseyrail City Line and had a timetable booklet in the same (12 hour) format as the Northern and Wirral Lines. It was only in the last few years that the Merseyrail brand has started to be phased out on it, however many stations still have Merseyrail on the signs. I suspect this was to reduce confusion surrounding the Daysaver ticket.
When did this happen? I don't remember it, and it's never happened on the Northern/Wirral Lines, e.g. they've never (outside disruption) spun trains at Maghull. The Wirral Line did at one point terminate at Hooton, but then that's technically outside Merseyside!
I travelled these lines regularly from about 1995, first "getting out to play" on the 50p Christmas special fares - they were certainly Wigan stoppers back then, they didn't spin at Garswood.
It's not politically difficult at all; the PTE would just add subsidy to the border.
Now you mentioned it, and got my little grey cells working, yes it branded 'Merseyrail City Line' (as well as Northern and the Wirral lines), that was certainly from the mid 1980's (if not before) until mid 1990's, as from about then, the new 'yellow' M station signs started to appear which had just 'Merseyrail' on them (bottom right corner of the signs). The latter stayed in circulation until the newer type without 'Merseyrail' appeared, with some depending on the line had for example 'Northern' on them from about 2010 / 11. By co-incidence, actually looked today (as a reminder) that I have a handful of station signs including a full set of the separate 'Merseyrail' Clty / Northern / Wirral signs.
The point I was making was that MPTE (as was) used to fund services from Liverpool as far as Garswood where they terminated and returned to Lime Street (presumably because they ran up against the MPTE boundary).
May be that was
before the 1980's? Certainly in the 1980's, the local service was Liverpool Lime St to St Helens Shaw Street (now Central) and return. In between that there were other services that carried on to Wigan NW and beyond were some stopped at Garswood and Bryn. The Merseyrail / travel boundary for trains on the north side of the Mersey has been Southport, Meols Cop, Ormskirk, Rainford, Garswood, Newton le Willows, Hough Green, and Liverpool South Parkway (ex Allerton) but the trains always operated beyond those stations anyway.