Close... My expectation is it isn't taking precedence for one over the other, it is configured to use one over the other.
When SQL was installed originally on the old server, the SQL instance name was "Myserver\myserver" (computer name\instance name). So when the old server was renamed to "Myserverold", nothing was changed on the SQL side to tell the services to look for "MyserverOld", so all of the services are still looking at "Myserver\myserver". So the services are trying to connect to "Myserver\myserver" not "localhost\myserver" or "MyserverOld\myserver" or ".\myserver" and "Myserver\myserver" is the NEW server.
There is nothing with "precedence" here. When you connect to the SQL instance, since it is a named instance, you would use the name "myserver\myserver" when connecting (unless you have a SQL alias set up) and the SQL browser service will pass you over to the proper port.
You can verify where the connection is coming from by looking at that CLIENT: section of the error. In your case 10.50.... is going to be the IP of the computer trying to connect to Myserver\myserver. I am confident that if you look up that IP, it will tie back to MyserverOld and not an end user machine.
The reason it is NOT in the old server log is that it is not failing to connect to the old server. Similar to if you tried to connect to the server myserver\myserver as sa but had the wrong password, you would not expect that to show up in the log for myserverold\myserver as you were not connecting to myserverold\myserver.