I don't think there's a single "correct" way to use that phrase, different people understand it differently. The meaning I'm most familiar with is an obscure exception demonstrating that your rule of thumb is usually reliable. For example:
"Street parking is always allowed"
"Not always! Three years ago there was a marathon that went through this street and you weren't allowed to park here that morning."
This exception is so specific and obscure that it "proves" (in a casual conversational sense) that "you can always park here" is a good rule. Not all exceptions prove the rule: if the exception is "except on weekends and holidays and overnight", that's so significant and obvious that it actually disproves the rule.
"Street parking is always allowed"
"Not always! Three years ago there was a marathon that went through this street and you weren't allowed to park here that morning."
This exception is so specific and obscure that it "proves" (in a casual conversational sense) that "you can always park here" is a good rule. Not all exceptions prove the rule: if the exception is "except on weekends and holidays and overnight", that's so significant and obvious that it actually disproves the rule.