After much enjoyable rummaging in the Scotland's People website, I can summarise:
There are several inconsistencies within records for the same people. For example, a second forename exists for both male and female entries connected with the Walton family, which is noted as "Gandy" or "Grandy" or "Gaudy". A similar problem also exists for the middle name of Hannah and this is manually recorded in summary details as both "Nore" and "More".
However, both the Birth and Death Certificates for Hannah state the name as "More" in the manually recorded summary details. And for those records the actual written entries do show an "M" rather than an "N" (or anything else). Also, Hannah was a witness to the marriage of younger sister, Constance, and the name is clearly "More" and not "Nore".
In all the cases of Hannah's name that I could find on certficates or census records, I tried to check the actual writing against other examples (in the same hand) to compare with other upper case "M" or "N" entries. In some cases, a clear example was not available but in others it was. One particularly good example was the "M" of "May" in the date of birth for Hannah (Sixteenth May 1862, at 2 hours 0 minutes am), which matches the "M" of "More".
So, for now, at least, I am confident that Hannah Walton, born May 16 1862, and who died unmarried aged 78 on October 10 1940, had the full name of Hannah More Walton and that it was she who painted and signed the decoration on my vase.