I don’t understand where “use the same domain name for my public IP” is coming, I never said I was doing this, you are talking about something entirely different now. (but, incidentally, yes it is technically possible to have two domains pointing to the same public IP)
If you know a way to do this, I am all ears. It is not possible with all split VPN solutions I have tested. (details of the VPN providers I tested are above).