Just update to the latest version  

What is the latest version? The page you referred to doesn't say. Mine is 6.0.0.120 - does the download give me a newer version? The one I have was installed only a couple of days ago.

The version you would get if you were to update now is only a minor version update (6.0.59.120), while it's always best to be on the latest version of Skype you should be able to link your Microsoft and Skype accounts using the version you have installed.

merge with your Skype ID as you sign up.

I don't see sign up anywhere.

If you have Skype set-up to log-in automatically then you might not have seen the log-in screen which gives you the option to log in with your Microsoft account (and go through the flow of merging your Microsoft and Skype accounts, which is what we refer to as the Sign up flow). 

To get to the log-in screen please open Skype, press Alt+S and click on Sign Out on the menu that appears. Then follow these steps to log in with your Microsoft account: https://support.skype.com/faq/FA12081

Sorry, I don't understand a word of this. What are my personal accounts? Or my professional ones? And again, what is the sign up flow?

This point is just to help users who have separate Skype and/or Microsoft accounts for professional and personal use. If you only have one Skype and one Microsoft account then you can just merge those two and ignore this message.

If I misunderstood the issues you are seeing then please let me know and I'll do my best to help you further.