ownCloud can be setup on your Linux machine. It supports a lot more than just notes (I use it e.g. for my calendars and contacts, so no Google here for that).
Then there are several apps supporting the ownCloud notes components, e.g. My Own Notes (also available at F-Droid) and MortNotes. I have not used the notes component with Android yet, just on the "Linux end": ownCloud runs as a web service, so you can access (and edit/organize) your contacts, calendars, notes, etc. from within a web-browser anywhere. The notes component supports Markdown, which not only gives you the possibility to well-structure your documents, but also to include/link to multimedia files.
To match this solution against your requirements:
- it allows you to take notes, to a degree with multimedia as well
- syncs without the need of a third party's server
- Supports Linux and Android, and many other OSses (as long as they have a browser and a network connection, you could at least always fall-back to that ;)
- The data format is open, and even better: it is Markdown.
As you write, you're "not interested in web-based note-taking software." I hope you don't mind if that is avaiable, too – as long as you have the other parts covered as well :)
I'm sorry I must pass on the Linux editor here. I'm pretty sure there was one, but never having the need for it, I don't remember which one it was. As ownCloud also supports, to a degree, OpenOffice/LibreOffice documents (read and write), I could name those; but you couldn't call them "light-weight"... As ownCloud supports WebDav, you could use that to access your files (including the notes), and use an editor like ReText – which offers a live preview to see what you type, PDF export, and more – of course, working with Markdown. I'm using this editor (and can recommend it by heart), but I've not tried to access ownCloud notes this way – so I cannot tell for sure how that works together (but am confident it does well).