Quotes are one of my favorite post formats, so we make sure this theme is handle it right.
The WordPress Codex defines the quote post format as the following:
A quotation. Probably will contain a blockquote holding the quote content. Alternatively, the quote may be just the content, with the source/author being the title.
As you can see, even the WordPress documentation notes this inconsistency in how quote content may be written by the user. Quote is one of the post formats supported by WordPress. It is used for quotations, specially when a user wants to just share a quote which is not within a standard post or article. A user may decide to add or to wrap a quote around
<blockquote> HTML tag.