Remember the days before smartphones (I'm thinking mid-90s here) where in the SMS settings of your phone, there was an option for "Email".
Anyone actually know what that did?
Did the phone actually send an email (as we know them today) from some kind of email account that your network provided for an extortionate fee?

I remember having a service where e-mails would be sent to my phone as text messages and I could reply to the text messages and it would send the reply as an e-mail. Think it was on one2one and cost £2 a month on top of my tariff IIRC

Yup, I used that on one2one as well

I'm sure there was also a option called "FAX".

Some networks (Vodafone) used to let you have an incoming fax number. Might still do.

I have a Vodafone PAYG SIM with one - so if anyone with a fax machine calls that number, the phone I use it with (Nokia 9500) picks up, receives the fax, and saves it for display later.

You can still do that. I sent an mms this week to an e-mail address as the phone could not be set up to an Exchange account I hold, plus it was not a smartphone. All you do is input an e-mail address instead of a mobile number. Not sure of the charge for this as the firm pays the bill.

Not ruling it out, but that sounds like a service which makes use of the existing SMS function rather than a service which would involve you having to flip to an "email" mode on your phone.

Hmm.. I seem to remember that, too!
I can imagine someone sending a text-based fax format (is there such a thing?) to a fax number, and the machine at the other end could just print it out. Whether that really is the case, I don't know.