At 2 my DD's absolute favourite was [[https://www.waterstones.com/book/you-choose/pippa-goodhart/nick-sharratt/9780552547086 You Choose!) which seems a bit more brightly coloured and with less text than the book you demonstrate.
DS was massively into Thomas books.
Agree with the pp saying that it's very hard to know if we don't know if he's just 2, 2.5 or almost 3.
Also agree that he will be expected to sit and listen if he starts pre-school or nursery, but they usually do that anyway.
I also don't think that the "That's not my" books are too young if they are happy to sit and look at them with you.
My DS went through a staff of not reading at 7 and I bought him the Beano, Football magazines and plonked the cereal box in front of him at breakfast. He kept reading and is expected to get good marks in his GCSEs this summer. I really don't think it makes much difference what you read as long as you read to them 
When the library reopens, I would take them and let them choose a few books, if you don't do that already 