Your second paragraph is true but so, I think, is what Sarah wrote.
Here is an "Original" image from my Flickr account.
Here is Flickr's "Large" version of the same image (which as it happens, is exactly the same size).
I am using Exif Viewer in Firefox. When I right click on the "Original" image and select "View Image Exif data" I see the Exif data. When I right click on the "Large" image and select "View Image Exif data" I don't see any Exif data.
If you don't have a "Pro" account at Flickr you don't have the "Original" size available to you, so you can't embed Exif data that can be seen here, at least not using Exif Viewer within Firefox.
I wonder, can you see the Exif data from the "Large" image above using Kuso exif viewer? (Reason I ask is that using Kuso exif viewer you can see the Exif data in Sarah's "Large" images over at Flickr, whereas using Exif Viewer I can't. (And I loaded an alternative, FxIF, and it said there was no Exif data available in the "Large" version over at Flickr.)