On the Merseyrail's City Line, most of the platform signage have either Merseyrail and some have Merseytravel. Merseyrail sponsor the services and Northern manage the stations, with the exception of Liverpool South Parkway.
Merseytravel is the Passenger Transport Authority for Liverpool and the surrounding area, they don't sponsor the stations directly.
Should the signs only say Merseyrail?
You are correct that Merseytravel is just the brand name of the Passenger Transport Executive or Authority of the Metropolitan County of Merseyside, which is just a consortium of Merseyside councils.
Merseyrail is just the private operator of the third rail (and now battery for the incredibly short distance between Kirkby and Headbolt Lane) electric services within this region, which does not operate the Northern services out of Lime Street.
Merseyrail operates every station it calls at except Chester and Lime Street (although the underground part there is staffed by Merseyrail) but this means it is the sole operator Liverpool South Parkway as well as Southport, Ormskirk, Headbolt Lane, Hunts Cross, Ellesmere Port and Bidston stations which other operators (in almost all cases Northern) call at, but nowhere else.
The name Merseyrail was created in 1971 by British Rail and applied only to the Wirral Line, before the lines out north out of Liverpool Exchange were named the Northern Line. Then its Merseyrail Project integrated these lines with Lime Street and extended the Northern Line south to Garston with the new loop and link tunnels.
It was even later than this, after sectorisation and moving away from blue and grey livery that many Regional Railways DMUs were painted in a Merseyrail yellow livery, in common with liveries from every other metropolitan area and used in a common pool nationally, although the Pacer refurbishment funded by Merseytravel included the large destination display which remained until withdrawal.
It seems it is only due to privatisation that confusion regarding Merseyrail became an issue, because Merseyside PTE had always wanted to use consistent branding with BR. Merseyrail Electrics (2002) Ltd which is the only Merseyrail operates its concession, which doesn’t include a “City Line” so the stations out of Lime Street shouldn’t say Merseyrail, but they do.