How do I make my RaspberryPi running Raspbian invisible on my LAN? Or, if that's impossible, at least customize what information is given from it?
The trivial way to do that is to shut down
If you want to listen to traffic, you can bring the interfaces up but don't assign any addresses to them. You can then run tools like
tcpdump on the interfaces.
If that's not what you want, you should be more specific with your question.
You can use the firewall
iptables to make the RasPi invisible to the network in fine-grained steps by adding rules to it. For example to suppress ping responses you can add this rule:
rpi ~$ sudo iptables --append INPUT --protocol icmp --jump DROP
Now if you try to ping the RasPi from another device it will not response.