How to modify a core theme and what is the correct way to do this?
What you want to do is create a subtheme of Bartik, and make your modifications to node.tpl.php there instead. Themes support the notion of inheritance, so this can be leveraged to tweak core themes without actually hacking core. As a rule, it is best practice not to hack core for the reasons you mentioned and a slew of others, security not the least among them.