The procedure is thoroughly documented in Codex under Moving a Root install to its own directory.
You misunderstand the point about changing URLs in it, and yes the terminology sucks. The only URLs that will change are those that are based on the “WP address”. That would be WordPress core, essentially the admin area.
All content URLs are based on “Site address”, following this procedure, that will still be the root of your site.
Since you want to retain uploads in place, you can do that by keeping wp-content in the root (splitting it out of core folder) and adjusting configuration for that. See Moving wp-content folder.
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