With all of the talk about new versions, I cannot for the life of me understand what is my next upgrade version from 2.0.13.2 that I am currently using.

I would like to get rid of it because most of the time when my visitors login they get a "why am I redirected here?" page. I have let them know that they may simply proceed because it doesn't mean a thing, but it is very annoying to new website people.

Should I upgrade to 2.2.4, or what?

The next upgrade version would be 2.0.14 when that becomes a 'Final' release - it's still at Release Candidate stage.

Upgrading to 2.2.4 is not recommended, and I doubt it would solve your issue as well.

Wait a little longer, as I believe RC2 - or perhaps the final - of 2.0.14 will not be very long before release... (days, rather than weeks...)