When I tried to open the pictures in another window (I am using safari as my browser), I was directed to sign into my flickr account. When I did so, I was told that the pictures are private and I do not have access to them.
So, I think you did do it correctly this time. However, you have to allow everyone to see them rather than making them private. Hope this helps in some way.
By the way, here's an easy way to post pictures without using links. If you do it this way, broken links is not a problem.
1. Open the picture on your computer screen.
2. Use the snipping tool (Windows logo key + Shift + S on a PC) to make a snip of the picture and temporarily place the snip on your clipboard. You snip by pressing the shift key while dragging the cursor over the area you want to snip. When you release the shift key, the area you highlighted automatically is placed on the clipboard and is ready to paste.
3. In the "reply" pane on TUG, make sure the BB code icon is black (= code is off). See examples below (images made with the snipping tool, BTW):
BB Code off (black - correct):
BB Code on (blue - wrong):
4. Ctrl-V paste into reply.
That's it!!! You should see your snip at the bottom of the pane. That means that your snip is there, ready to go. When you click "Post Reply", your should see the image embedded in your post.
If you get a message after you paste that the image is too large, go back to step 1 and zoom out to reduce the size of the area you want to snip. Then repeat the process.