So it's a matter of mere preference, not current law? That is, that there are not certain things that MUST be passed by 60 votes and that can't be passed by 51 by overriding the filibuster rule? Something like: it can only be done X times per Senate session, or X times for type Y legislation per session?

The Senate rules currently require 60 votes to close debate on nearly every matter that comes before them. There are specific exceptions for certain things: many confirmation processes and reconciliation votes, primarily. Everything else requires 60 votes to invoke cloture - the formal term for ending the debate process and ending a filibuster. The budget is not something that is exempt from the rule. So it would take 60 votes to end debate on a continuing resolution.

The Senate rules are binding on the Senate, but they can also be changed by the Senate. Until they are changed, the Senate has to comply with them.

The GOP is loathe to eliminate the filibuster for budget matters - and the Democrats don't actually want them to do so. These types of changes are colloquially referred to as the "nuclear option" for a reason. So although the Senate could amend their rules to allow the CR to pass with only 51 votes, that's not at all a likely outcome, which is why it doesn't get talked about too much as a possible way out of the shutdown.