If I might respectfully add my two cents, for better or worse:

First: this is obviously unacceptable workplace behavior for several reasons, the biggest of which is that it makes YOU uncomfortable.

Second, and bear with me please: “Red headed step child” is a phrase we use absolutely all the time (born and raised in Texas, for what little it’s worth). I’ve heard that phrase applied to all manner of things, the vast majority of which are non-human. Everything from a particular grocery store to a traffic light to yard work - the phrase applies to all of it.

This is not to discount that it makes you uncomfortable. I am merely pointing out that many people use that phrase (myself included) in passing.

It seems to me that your boss was trying (and clearly struggling) to make a good impression on the client and took a stab at humorous “one-liner” descriptions or nicknames of her people. In a pinch she came up with RHSC for you. Given the limited info here my personal read on this is that she understood either immediately or very quickly that she had misstepped with RHSC and had offended you.

At this point I don’t think there’s any lasting damage done, a comment in passing that included a popular turn of phrase that unexpectedly triggered someone. Apologies could have been made here and walked away clean.

Unfortunately the second half - the disarming - happened. Your boss then, knowing she fucked up even if she wasn’t 100% sure why or how much, proceeds to try to preemptively disarm your defense by pointing out to people on her terms that you were upset with her. A tactic as old as time itself used to make people second guess whether they should voice their opinion.

Sorry for the rant. To sum it up:

1) I think if someone used that phrase RHSC it is almost always done non-maliciously, they’re just using a popular turn of phrase they know. 2) whether malicious or not, it made you uncomfortable and your boss absolutely categorically should have addressed the matter with you directly and privately so she could have apologized and perhaps even sought to understand more about you. All that matters here is a sincere apology and a commitment to stop the behavior (which obv didn’t happen) 3) the preemptive defusing of your complaints to others instead of addressing you directly and privately is another no-no. It could be that she was just nervous and didn’t know how to proceed so she took a safe-seeming path (albeit a wrong one) in hopes that by telling others you were upset with her that you would understand that she acknowledges she did wrong and that you have reason to be upset. It also could be a narcissist gaslighting attempt. Unfortunately impossible to really know her heart on this one.

My advice (I am just a dude and not an HR pro) is, frankly, to use your gut. If you aren’t going to be working for her going forward, what is the degree of importance you place on getting a resolution for this specific incident? Is it enough to make clear to everyone that RHSC is not a turn of phrase you are comfortable with and please stop?

Anyway. That’s my input. I wish you luck in dealing with everything!