Email server settings

Almost certainly the REAL name of the server - its so called canonical name - is mail.bobsserver01.co.uk And your domain and website is aliased to that via a cname.

So either will do when collecting mail via IMAP or POP or sending it via smtp.

probably it looks up the mx server for your domain

because they both resolve to the same IP address 5.77.60.195

who edits your DNS? If it's them, and they keep mail.yourdomain up to date with their moving smtp server, should be fine.

mail.ourdomainname is probably an alias for one (or more) of their physical servers. One level of indirection won't hurt at all.

The advantage is that it should redirect if they get taken over by Cate.

mail.ourdomain.co.uk is a "CNAME" for the apex domain ourdomain.co.uk

ourdomain.co.uk has an "MX" record of mail.bobserver03.co.uk

the host mail.bobserver03.co.uk and the apex domain ourdomain.co.uk both have "A" records 5.77.60.195

obviously you can't use a CNAME in an MX record.

But your mail software might warn about an invalid security certificate when you use mail.ourdomainname.co.uk

If you collect email via an unencrypted connection (say IMAP on port 147 or POP3 on 110), then both are fine - it sounds like your DNS records contain a CNAME record that makes imap.ourdomain.com or pop.ourdomain.com to the ISPs mail server(s)

If you use a TLS encrypted connection (usually ports 993 (IMAP), or 995 (POP3)), then using the CNAME version may get you a certificate warning when you connect since the domain name contained in the certificate will be that of their server and so not match your domain name.