The RHS book sounds confusing but it is simple really.
The first para you quote is for summer raspberries, and you have indeed removed the old canes (which would have fruited) as you say in second para.
Good that your autumn rasp’s have all produced new leaves, don’t prune them as they should give you fruit by August and September, even if you planted them this spring.
Canes which I planted last December (autumn fruiting) are half dead from the March ice, so you are doing well. I think we had a milder January and February here, encouraging early bud swelling which then got destroyed in March. Steady winter cold is better!