No you don't. Essentially the ticket is a barcode and can be presented on any media you choose be it a piece of paper, smartphone screen, laptop etc. Similar to how tickets work for many concerts and sporting events these days.
I have noticed that the LNER website says you can download e-tickets to your Apple or Google Wallet and show the ticket on your phone, or print it out, but it doesn't mention that you can simply have it emailed to you as a PDF and show it on your phone.
Do you know if this means it won't be valid if you show it as an email attachment? I know some conductors won't accept screenshots of tickets, but that's not the same as showing an original PDF.
Also, if you want to have your ticket emailed to you as well, you have to tick the box asking to have it emailed. I think most other train operators that offer e-tickets, such as East Midlands Railway, do this automatically.