Is it possible for User2 to access the local SQL Server without reinstalling?
Here is the error:
Cannot connect to mydatabasename.
Login failed for user 'myuser2name'. (.Net SqlClient Data Provider)
Server Name: mydatabasename
Error Number: 18456
Severity: 14
State: 1
Line Number: 65536
For help, click: http://go.microsoft.com/fwlink?ProdName=Microsoft+SQL+Server&EvtSrc=MSSQLServer&EvtID=18456&LinkId=20476
The easiest way would be to log in via the "sa" account but I expect you don't have the password to this or it isn't enabled.
If you start the server in single-user mode you will be able to add additional users without reinstalling.
This MSDN page outlines all the steps but in brief they are