Based upon your hosting provider's recommendation (specify server, port, username, and password), SMTPAUTH should be used. Do not forget to fill in all of the relevant SMTP settings.
Would also recommend "send from known domain = true".
One could also temporarily enable email logging to verify emails are sent by Zen Cart. Keep in mind the email server (not Zen Cart) is ultimately responsible for dispatching and delivering emails. Very similiar to sending a postal letter (Zen Cart = You; Email Server = Post Office).
PHP = Rely on hosting provider's configuration of php.ini to tell PHP how to send email. Does not use Zen Cart configured username, password, server, or port. Does not usually provide any feedback regarding if the email was received by the email server.
Kinda like placing a letter in the office "outbox" and trusting it will get to the Post Office (eventually and by someone else's hand).
SMTP = Use SMTP with no username or password sent to authenticate. Does use Zen Cart configured server and port. Does not use Zen Cart configured username or password. Will report if unable to contact email server and if email server reports an error while receiving the email request.
Kinda like placing a letter in a Post Office box (with no verification of your identity or return address) and trusting the Post Office to deliver the letter (even when / if you are pretending to be someone else on the envelope).
SMTPAUTH = Use SMTP with Zen Cart configured server and port. Send Zen Cart configured username and password to authenticate. Will report if unable to contact email server and if email server reports an error while receiving the email request.
Kinda like going into the Post Office, having them verify your identity and return address, and then trusting the Post Office to deliver the letter.
As for those error logs, I would "guess" you are running an older version of Zen Cart (and / or modifications / pluguns) on a newer version of PHP than supported for the older version of Zen Cart. One can determine the version of PHP and Zen Cart by checking the "version" link in the Zen Cart administrative interface. These can be checked against the Zen Cart requirements documetation.