"Cannot open user default database. Login failed" after altering MDF and LDF locations and bringing DB offline
I may have input the incorrect Name parameter above (Would the mdf/ldf files be dropped in this case?)


This would not have dropped the files, if the NAME values weren't correct, the command would have failed.


  I also did not do step 2 in the answer referenced above (Stop log reader agent, distribution and merge agent (if they are running).


Are you using replication on this database? If not, then this doesn't matter as these steps are not necessary.

When you login to SQL Server via Management Studio, you can explicitly select a database to connect to or use your default database. If your default database is unavailable, you will get a login failure.

This link details how to explicitly select a database to login to in SSMS. Basically, on the login window, click Options then select to Connection Properties tab. Under "Connect to Database:" type "master" and try logging in again.

If this allows you to regain access to the SQL Server, follow the steps in this SE answer to change the default DB for your administrators to "master" so that they have a better chance of being able to log in even when user databases are taken offline.