Is it possible for Websites to define the Image/Title - or even the whole Aadaptive Card - to be shown in the Windows Timeline when they are added by Edge.
Yes, it looks like. Not sure about all the rules - but just by reverse engineering a bit on an image I saw on my Timeline history contributed from Edge for a random page like https://github.com/metabase/metabase/blob/master/test/metabase/test_setup.clj
If you inspect the source code for that page you'll see:
<meta>
...
<meta property="og:image" content="https://avatars3.githubusercontent.com/u/10520629?s=400&v=4" />
...
<meta property="og:title" content="metabase/metabase" />
....
That corresponds to what I get rendered as background image and title on the Timeline card. The og:<something>
meta properties are defined by the Open Graph protocol: http://ogp.me/