It sounds cost effective to use the domain forwarder of registrar to redirect to another domain rather than using your server. In practice, I have found that the domain forwarders from many domain registrars were never as good as htaccess directives or nginx directives.
They may block googlebot:
Depending on the user-agent they may block useful bots like googlebot. You may not aware of it and you will have not options to check it since google webmaster console does not accept DNS records as verification of property.
They may block IP addresses from bots:
Once again, they can block IP addresses from bots without you knowing it. As I said before, Google webmaster console does not allow website owners to confirm their ownership using DNS records. Consequently, when a domain is parked in the server of the registrar with a redirection in place, there is no way to check if googlebot can crawl the old domain.
The service may not be as reliable as you think:
The redirection may be down from time to time without you to be aware of it. It is a free service so there is no pressure to make this service reliable for the registrar.
Directive files are more reliable:
It is the most likely scenario. htaccess or nginx directives are more reliable and leave less room for errors than domain redirections from registrars.
I have checked the http headers response and the html response of domain redirections from registrar. I have discovered that some of them were not in compliance with RFC 2616. Some added unecessary url in the html of 301 response that pointed to another website that wasn’t mine. In the http headers there were my target url, but in the html, another target url was displayed. I have also noticed that sometimes, a www redirection didn’t work while the non-www worked because I forgot to add a line in the redirection tab from the registrar. All of that led me thinking that I took a risk for nothing as it didn’t cost anything to add the redirection to my server.
Advice for your domain redirection:
Since your domain was hosted by a webhosting company and if it had some ranking it means you had no issue hosting this domain on this server. Consequently, it would be logical to host the redirection from this old domain to a new domain on this webhosting account.