No ...You should trim the thin spindly canes down to the ground whether they are summer or autumn fruiting. The big new strong growth should be next years summer fruiting canes.. if they are anything like mine they will still be a bit green and bendy and there will be no sign of fruit being formed on them. The autumn fruiting canes will start to look dry and brown, mine still have a few flower buds on them and even a couple of raspberries (they won't make it to ripeness I fear). Those are the ones that you need to cut back to ground level. If you are still not sure (and apart from getting someone to show you I don't think we can add much more) then pick out the best strongest canes out of all of them and tie them in and cut the rest back to about 18" high.
The autumn fruiting ones will put up a small crop early next year and you will see shoots starting at ground level very early on. The summer fruiting won't throw up canes until later in spring. Once the autumn one have fruited in spring and there are are canes growing alongside cut the autumn canes from this year back to the ground.
Next year get some red and blue insulating tape and clearly mark the summer and autumn canes so you know which is which next autumn.