I just did a quick search on the Reblog concept because, admittedly, I've never used Tumblr and had no idea what you were talking about. After reading a bit about it, it started to sound very familiar ...
Most blogs today allow you to add a ReTweet (Twitter), Like (Facebook), or Share This (Generic) button directly on the post. These are very network-specific and key in directly to the APIs of various social networking and crowdsourcing applications. Tumblr, since they run a succinct network of blogs, did the same thing. The Reblog button allows you to quickly repost content on your own site within the Tumblr network. As one site mentioned, not very extensible, but still pretty cool.
The difference is, each self-hosted WordPress blog is its own web application. So there's not really any way you can have blog authors put up a "share on your own self-hosted application" widget on their sites. Fortunately, there is a solution.
create a post by quoting some text, images, and videos on any web page.
To use Press This, go to your Tools sub-panel and drag the "Press This" link into your bookmarks toolbar.
When you're browsing the Internet and you find something you like, highlight the text (or don't and you'll use the entire page's content) and click the Press This bookmark. A pop-up window will appear allowing you to instantly edit and post the content you're viewing.
As far as showing up in the notes section of a Tumblr blog, that's an entirely different issue. You're looking for a very specific, custom integration between two different platforms ... I have found one plug-in that will automatically post to a Tumblr blog from WordPress, but the last time it was updated was April of 2008 so I'm not sure it'll be too much help. There's also a synchronizing system called Tumblrize that will automatically post your WordPress content to Tumblr, update it when you change things on WP, and update the WP copy if you change things on Tumblr.