Not sure of some of the names on the Viking map ( underlined in red) as to later map.

Just some of my guesses.

Meir: Meols
Porsteinnstun: Thurstaston
Langholmr: Leasowe
Klakkrtun: Claughton
Geittun: Gayton
Ærai: Arrowe
Habyr: Barnston

Can't see the map but when you bear in mind a 'th' sound would be shown by a kind of pointy P, I reckon the Thurstaston one is right.

That would make sense because Thingwall is þingvollr.

It has Kirkjubyr in the Wallasey area which means Kirby. West Kirby is, Vestri Kirkjubyr, so we know Kirkjubyr is Kirby.

I think Kirkjubyr means church or similar so Vestri Kirkjubyr may mean church to the West
kirk
Scottish and northern English, and Danish: from northern Middle English, Danish kirk 'church' (Old Norse kirkja), a topographic name for someone who lived near a church.

Vikings arrived on the shores of West Kirby after being expelled from Ireland and West Kirby became known as 'vestri Kirkjubyr' which translates to 'West Village of the Church, possibly St Bridget's Church

If it translates as 'West Village of the Church', West Kirby could possibly be the village west of Wallasey, known as 'the Church'. Are there any possible locations of a long lost church in Wallasey?

Lingham is Lyng-holmr in Old Norse - heather island on a marsh.
The map shows Langholmr.

Eskeby (in Bidston), Warmby (in Heswall), and Stromby (Thurstaston).

https://www.liverpoolecho.co.uk/news/liverpool-news/merseysides-viking-history-z-everything-11031508

Below is what Wikipedia says about St. Hilary's Church - given the prominent position, it has the potential to be a very old religious site, definitely back to the Saxon era and possibly beyond? If you ever have the chance to see inside the tower (Heritage Days etc), there's a rather smart Minton tile floor in there.

'The present building is believed to be the sixth church on the site.[4] The first church is thought to have been built in Saxon times and was probably built of timber and of which there is no trace. Several stones have been found of a Norman structure on the site. It is thought that a new church was built between 1162 and 1182 by William de Waley. This was rebuilt and a tower added during the reigns of Edward I and Edward II. The next rebuilding was in the age of Henry VIII when the tower of 1530 was reconstructed. In 1757, the church was described as ruinous and it was rebuilt.[5]

This structure was accidentally burnt down in 1857, reputedly because the sexton over-stoked the heating boiler, and bacon being cured in the boilerhouse began to drip fat, causing the old prayer mats beneath them to catch fire. Because Wallasey had no fire service, a messenger had to be dispatched to Birkenhead, and by the time the fire brigade arrived the structure was beyond saving.[6] The church was rebuilt as a separate building in 1858–59, leaving the tower as a freestanding edifice. The architects of the present church were W. and J. Hay.[7] '