Just to clarify ... are both sites using the same provider?
phpMyAdmin is usually tied to an account with the provider.
So to migrate a site to a different provider, one would need an sql dump (can do from the phpMyAdmin) on the site that your are moving. That saves to an .sql flie.
Then in the new site, if phpMyAdmin is available and running, one imports the sql dump from the site to be moved.
If sites are hosted with different providers, neither provider is responsible for porting your sql dump to the new server. That's customers job.
Still ... think you've missed a question asked ... the new site looks like it already has a Moodle running there - just not in the directory you gave. Try the second URL (the one you are moving to) without the /directory and see what you get.
'spirit of sharing', Ken