you can really tell the difference between an angry at leo Raph/angry at my brothers raph vs a raph who looks at you like your dirt beneath his shoe or really a enemy that needs to be beat. Also Raph may act that way when it comes to his brother leading, but its way worse if someone he doesn't care about or even respects as much as he respects leo tries to give him orders. Especially if Leo's not there to tell him they need to work with this person.
And thats not even getting into Leo's reaction to anyone insulting his brothers and how they act as team.
You said it!
A Raph who is angry at the world (or a villain) versus angry at his brothers shows a coldness that doesn't exist in him otherwise. You can actively see him becoming a machine who thinks of nothing but Revenge when he sees one of his brothers taking a hit, or when he saves them or is desperate to find them.
Those times when he has a family member to actively defend, protect or avenge are one of the few moments Raph allows himself to forget everything he's ever learned about controlling his emotions and lean into that raw rage that he knows will get the job done, because he's strong and fast and unstoppable.
Raph's ability to turn that off and on like a switch is why his brothers in 03 refer to him as a Secret Weapon. He is their last line of defense. He is the guy that will not stop until he's put down for good. (Is it really any surprise that his character's origin was inspired by X-Men's Wolverine?)
It's no wonder that Leo finds Raph to be the most challenging brother to lead. His strengths are without measure. It has to be hard to know when and where to flip Raph's switch.
But Raph's "anger" at Leo is always tempered with respect. He has high standards for Leo because Leo has set high standards for himself and for the rest of them. He's never going to miss an opportunity to remind Leo that it's all on his shoulders and he better rise to the occasion. (And Leo is pretty good at doing that, despite all of his own fears.)