We know that direct deposit is a type of ACH. Is there any benefit or difference between setting up direct deposit vs the payer executing an ACH push (i.e. time for payment to arrive, costs, etc)?

There is essentially no benefit to you. The benefit is for the end-customer or the payroll provider, by having a direct deposit agreement they are allowed to send recurring credits to the same account with customer authorization.

The main key benefit is reliability, and also the funds via payroll are considered “safer” since the payroll companies take the money from the employer a few days in advance. A random push of money from an employer is considered more risky.