RE: My other option is to upgrade to a virtual server host package or, most likely to use a dedicated email service provider like SendGrid
I would suggest going straight to the SendGrid solution. The basic setup is rather quick and easy. The SPF/DKIM is optional but does help a lot in preventing email from ending up from being blocked or filtered as spam. Going the virtual server is moe of the hammer approach. And, even then, having a dedicated email provider is still a better option.
RE: I’ll try to figure out how to get rid of the “website” field of the Contact form (I think that encourages spam) and change it instead for a “phone” field, with would be a lot more useful for my purposes.
Different topic??? The contact addon does not have a website field.
Hope this helps!