Might sound daft but to test which colour the icon was on the mobile app (green I think) I sent myself a message.
Whilst the message does display under Sent, it does not appear within Inbox, nor is a notification generated and as such, no notification is displayed in the mobile app.
Not tested on desktop but assuming the same outcome.
I am not sure if this is by design, if it is, perhaps the To field shouldn’t allow the selection of oneself, preferably though, the message would appear within Inbox and notifications generated etc so that the behaviour is the same regardless of recipient.
Just tried tagging myself within the message and that doesn’t change the behaviour above either. No notification for a mention.