if I want to book a train, let's say, from Amsterdam to Brussels or from Brussels to Cologne, can I buy a ticket without need to show my EU card and hence without being registered in a system? And if I can, are there police checks that register my identity?
Train travel is essentially anonymous. If you buy a ticket at a manned ticket counter, or from a ticket vending machine you will not be asked to show an ID, nor will you be required to show an ID when boarding a train.
The railways do not care who travels on their trains. Trains do not have passenger manifests.
They do care that everyone on their trains has paid for the privilege. So they do care that your ticket is real. With tickets printed at a ticket office, or in a ticket vending machine this is not an issue, as these tickets are printed on real ticket stock. However with print-at-home tickets there is the real risk that people would start photoshopping their own tickets, or printing multiple copies and sharing them with friends.
So for print at home tickets a name will be asked. This does not mean you have to register (although some railways encourage this), but it does mean that you need to have a piece of ID with you (as you need in most EU countries anyway). The purpose here is to avoid counterfeiting and copying. The name you give when booking is encoded in the ticket, in a way that prevents you from creating the ticket yourself. And because your name is on the ticket it is only useful for you.
But the purpose is not to keep track of your movements. It is just an anti fraud measure.