It's NTN 6204z on upper and NTN 6206z on lower. I can't say which one
is rattling. I'll replace both.
What kind of puller do I need to remove the bearings? Do bearing
dealers provide any tool rental or I'll have to buy a puller etc.?

Before you dig too deep, look up a document someone here on
RCM wrote called, IIRC, "How to buy a drill press" and use the
checklist he has to see what's actually wrong. As I remember,
bearings are not the only parts of these that make noise, it can
also be worn splines.

Hope that helps,
--Glenn Lyford