Is the number of edits at http://stackoverflow.com/users?tab=editors&filter=all the one that counts the edits relevant for gaining the Strunk & White and Copy Editor badges? (If not, wouldn't it be nice to have it that way? There's no reason for two different edit counts named the same.)
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Nope, they're not the same, for two reasons:
No, the edit count shown on "Editors" tab could be different, I think it also includes "retags". To get an edit count that are eligible for "Strunk & White" badge, try running this query on data.stackexchange.com. Note, data.SE will not give you the latest count as the data is dumped every month:
"Strunk & White" is for those users who try to fix spellings and grammar of question titles and body.