Edit function in ‘Visit Site’ in wp-admin WP5.0.2

I recently updated to WP 5.0.2. In previous versions, when I used the Visit Site option in wp-admin there was an Edit option for the page visited. In version 5.0.2 on zeromost.byethost4.com, there is no Edit function when the page is displayed. It is counted by the counter plugins (like WP-Statistics) as if a 3rd party visited the page.
This behaviour is not consistent. On another site of mine (hardlynatural.byethost7.com), also on WP 5.0.2 the Edit option is displayed when I used ‘Visit Site’ via wp-admin.
I can’t seem to locate what’s making the sites behave differently for that function. I’d like to restore the Edit option for the first site.
Any suggestions much appreciated.

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