I selected Move/Copy Fields and looked for a "Last Played" in the list shown in the Move/Copy Field dialog box but it was no longer there. Only my new field (Last Date Played) was in the list.
It would never show up there. Probably because its Edit Type is "Clear only".
Normally you would have created the "Last Date Played" field in "Tools>Options>Library and Folders>Manage Library Fields" as well. Did you create it by adding it to a view?
Anyway, to get the contents of "Last Played" into "Last Date Played" just highlight the files you want to copy the date across for, then edit the "Last Date Played" field (rename), and insert the expression "=[Last Played]", without the quotes.
Note that the data will only be copied across if there is a date in the "Last Played" field. If the field has "never played" in it, that won't be copied across.
BTW, the "Last Played" field is automatically maintained by MC based on when you actually played a file. I understand if some files don't have the data and you want to have the information that you know in MC. But you are making a lot of work for yourself to maintain that field. I don't know of any way to update the original default field though.