Difference between "rule" and "law" in scientific context
The only way I've ever heard rule used in a scientific context is with various conventions for solving problems — eg. Right-hand rule (for finding cross-product directions), Kirchhoff's Rules, etc. These are human constructs, but I suppose they're based on phenomena in nature that we've repeatedly observed and that seem to hold true. (so perhaps you could consider them as synonymous to laws).
According to livescience.com, a scientific law is:
"The description of an observed phenomenon. It doesn't explain why the phenomenon exists or what causes it."
As a separate note, it's important to recognize that one can't "prove" a scientific statement, and so no law or rule can be "proven." Science is based purely on making and attempting to explain observations of natural phenomena, and phenomena can of course change at any time and go against long-held observations - we simply don't know enough to make rigorously proven statements about nature.
There is the strict definitions and there are the coloquial uses (see above). The strict definition comes from statistical Physics (and therefore is valid for all sciencies). In short: laws - define interactions/behaviour while rules - define distributions/physical properties. Therefore, laws are deduced from hypotesis and their verification/falsification and rules come from directly/indirectly but measured physical/any properties. Rules and laws are mutulally exclusive and a given statement cannot be both a rule and a law.
For simplicity outside of certains fields of physics (like statistical Physics, general relativity and all quests for unifying theory) more fluid definitions are used despite not being strictly accurate.
As for theorems, postulates and formula they deal with different types of categorization and are not mutually exclusive in use with rules and laws.
I think they are synonyms when used in a scientific context:
Law:
a general rule that states what always happens when the same conditions exist:
Newton's laws of motion,
the laws of nature/physics
(Cambridge Dictionary)