Indeed they do. And taking it further, ‘single’ suggests never married, whereas ‘divorced’ does. But knowing human nature and having experienced the female of our species in the context of marriage a couple of times, I would not expect one who initiated the divorce to identify herself as ‘divorced’, as it implies, subtly, that her husband divorced her. Thus, ‘single’ could easily mean ‘never married’ or ‘I divorced him’... Sigh...
A mess all around.
Or after 30 or 30 years si gle a better fit than divorced