…the option to copy a video URL with a specific starting point.
Previously when you clicked the Share button beneath a video paused at a certain point you were presented with a URL which included that start time.

Now that option is gone (although it’s still there in the Embed option):
That the box is titled ‘Message on YouTube’ seems nonsensical also.
The good news is you can still copy a video URL with the time point in by right-clicking on the video:

Why the first method has vanished I don’t know. Intentional perhaps, lost in an “upgrade”, possible.
Alternatively, copy any video URL and include ?t=T at the end (where T is the time in seconds from the start).