Zaniax

Lewis - the cause of these error messages is one and the same.

You cannot connect to your IMAP server via an 'external' router with an different ISP without info about the SMTP server used by this 'external' provider.

The name of the SMTP server changes (normally) with the Internet System Provider (ISP) despite of the mailing setup of YOUR 'private' ISP (POP3 or IMAP)

The 'external' SMTP server cannot recognize the name of your 'private' SMTP server and therefore it returns a error-msg saying something like:
'I cannot recognize the recipient of your massage (because I cannot recognize the name of the SMTP server you gave me)'

Its not exactly what happens but the end-effect is the same.

Again, use a web based email provider and eventually redirect your emails automatically.
You may still retained your 'private' email setup to be used when you're at home (if you prefer so) and select the 'external' setup when you're out somewhere.

Personally I use the web based setup all the time after I retired from my job.

Email systems are complicated - but hopefully you're a little more clarified now :-)


Edit:

Just to make myself clear:
You can use a web based email system via your normal email program -
that is, you don't have to access a web site in order to use emails.

I use the built-in email program of my x3i to access gmail.com
The emails appears on my x3i and are still retained on gmail (despite using POP3)