what is the difference between a "push" and a "transfer"?

Great question!

A push is an internal account change at a registrar. If you purchased a domain name from me that was held at dynadot I would ask you for your email address and dynadot username. I would login to my account and use those details to push the domain name from my account to yours. Some registrars will require the gaining user to approve the account change while others will not.

A transfer is an external account change and is also a registrar change. If you purchased a domain name from me which was held at Dynadot I would first unlock the domain name and then send you the "auth code" for the domain name. You would login to your gaining registrar and submit the domain name and auth code. You will probably have to pay a typical registration fee for this and most registrars will add one year to the expiration day once they receive the domain name. I have seen transfer take from a few minutes to a week to complete.