It really isn't that tough.
I'll try again.
There are two different products with the same name and different benefits.
TfL - Transport for London - the folks who run the Underground, the TfL Rail (the slower stopping train between Heathrow and Paddington, and another commuter train in the northeast of town), the buses and trams - issue a product available on an Oyster and available on a paper ticket with the Roundel symbol (the symbol of Transport for London, a circle with a horizontal bar across) in the corner. Travelcards with a Roundel on it and Travelcards on Oyster are valid on all Tubes, trams, virtually all buses and many trains within Zones 1-6. Travelcards on Oyster and Travelcards with a Roundel are not valid to obtain the 2 for 1 discounts on the Days Out promotion of the railways. TfL Travelcards with a Roundel or on an Oyster do not require any further ID and no photo is required. They are available in one day, one week, one month or one year lengths.
Railway companies (there are around 26 of them around the country) issue credit card sized tickets for riding trains. They can be from station to station or from a station to a London Zone(s), or simply for 4, 5 or 6 London Zones (one day), or for 2, 3, 4, 5 or 6 Zones (longer duration). Within the names Zones on the railway Travelcard they have the exact same validity on transportation as a TfL Travelcard. Railway issued Travelcards are not available on Oyster, they are printed on either normal train ticket stock or on a special Travelcard ticket (depending on the facilities available where you get them) and are identified by the "Crows Feet" or "Double Arrow" logo. Train company Travelcards will not only have a Roundel. If you get one at certain dual stations the ticket may have both the Roundel and the Crows Feet. The Crows Feet logo identifies it as railway issued, and it must have that (see the details in the FAQ which I actually thought you understood a while back - and look at the pictures again) to be valid to give you 2 for 1 at the deals offered by the Days Out promotion. A one day rail company issued Travelcard with Crows Feet does not need any further ID nor a photo. On the railway any ticket with a validity length of 7 days or longer is considered a "Season Ticket" and all railway Season Tickets require a special card to accompany them with your name, your photo and a long number on it. The ID card is free - all you have to do is provide a passport sized photo which is a good likeness of you. It could be a print from your phone as long as it is the right size and easily recognisable as you. You bop up to the window and say to the chap or chapess behind the glass that you would like a 7 day Zones 1-3 and a photo ID, Please. And you hand him or her the photo. They ask your name and laminate the photo onto the card and ask for the money for the Travelcard. You can tap your phone or contactless card, or stick your card in and PIN, or swipe your card and sign or hand over money. You won't have to show your photo-ID except on trains, but you have to have it.
So - your choice.
If you don't want to have a card with your photo on it or you don't want the hassle of a couple of minutes at a window in a station, and you are willing to forgo the benefits of 2 people paying for 1 at some attractions, just put the Travelcard onto your Oyster.
Or find a photo (one each), use a ticket issued by a rail company, and enjoy the benefits.