Does anyone know if the Stripe payments count as a direct deposit for purposes of new banking client bonus?

That would depend on the terms of the bank. Ask the banks directly about funds transferred to you as an independent contractor/small business.

As Deb said, you have to ask this bank. Generally direct deposit is only viewed as a regular, ongoing payment from an employer. These employers generate payment files that go through the ACH system. Rover is not your employer and your deposits would be irregular and of varying amounts.