When the user logs in using the Login activity their information is checked in a database. A response is then send to the Login activty telling the user if their login attempt has been successful.

When a new user registers, the data they enter is validated in the Register activity. Their information is then send to the database. The database will then send a response letting them know if it was successfully entered.

Is this the correct way to illustrate these actions in sequence diagrams?
The vertical lifelines typically do not represent activities, they typically represent instances of classes. (Yes, it is syntactically possible in UML to use an activity, but it is a very strange thing to do.) Those instances are usually typed by classes or interfaces that either have operations or signal receptions that can be invoked via messages. Moreover, in UML, there is no message kind that has a white arrowhead.
Therefore, no, those diagrams are not quite the correct way to illustrate the actions in a sequence diagram.
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