If the IOM was the same as England there was no legal requirement to register a change of surname. You could call yourself anything you liked provided it wasn't for a fraudulent purpose. This existed until enemy aliens were forbidden to change their names in 1916, and the rule extended to all aliens in 1919. However people found it inconvenient to have no formal record of a name change and often formalised the change in various ways including by Deed Poll, advertising in the local press, by Royal Licence (when a condition of a will), etc.
I very much doubt that a minor change such as Moore to Moores would be recorded anywhere.

Sue