In my family and circle of friends every new baby is celebrated with a shower. I have a cousin who has four children and had a shower for each.
Men and women are usually both invited and we see it as a chance to catch up, celebrate a new addition and shower the mum to be with love and yes, with presents too!! It's just what we do.
No one sees it as "present collecting" only as a super nice reason for getting together, being a family and showing mum to be that the new bub is just as anticipated as any other.
I would absolutely have a BBQ for number 2 OP... And call it anything you want!!