Does renaming a page's path automatically redirect from the old page path?
Yes, page alias are added when a page's path is changed.
Pages can have any number of aliases. When a page's path is renamed the system will automatically add an alias from the page's previous path to the current page.
This prevents 404 errors which are a well known cause of user frustration and lower SEO rankings.
When a user requests a URL first a matching page is searched then the page aliases are searched, if a matching page alias is found then that page is used. There are NO 301/302 errors returned there is just many paths to the same page.