"There are the neighborhoods with Hispanic people but a lot of them don't speak English and the Blacks who don't trust us. And then there's the normal neighborhoods, which are easy."
There are a lot of different ways to interpret this. For example, the "abormal" thing could be that people there don't speak the language of the country (ie, English), or don't trust members of your community organization... None of these have anything to do with race issues. It could be abnormally high crime, or poverty, etc.
We are both "white" and her implication that non-white neighborhoods are somehow abnormal really bothered me and I had to step away from her because I didn't know how to respond.
That's unfortunate, as the best thing to do would have been to say something very simple like "how so?" and let her explain and elaborate her way of seeing things. Since she's an old member of the organization and you are a newcomer, a conversation where she gives you all the information you want will flow naturally. In fact this conversation would occur on its own unless you actively prevent it, for example by going all out based on one single word which may have been a misunderstanding, and then leaving before you could have all the information...
In other words, neither you nor anyone here has any idea what she actually meant. It's all assumptions.
I know that she doesn't mean to be casually racist because that goes against the purpose of this organization and our outreach but I'm not sure how to respond to her to recommend a better phrasing than the white = "normal".
Yeah, if she spends lots of hours every week helping poor blacks you can be pretty sure she isn't racist, and she probably thought your reaction was silly. You make no mention of her reaction in your question, which is unfortunate, as it would be useful to know what she's thinking... but I'm sure she noticed you cringe when she used the word "normal", then when you awkwardly exited the conversation. So, just assume she knows exactly what you think and is most likely wondering how to talk to you about this and fix the issue. The whole situation is so cliché it would be hard not to.
So, you can try something like this:
"You know how people of my generation are always told to be careful not to sound racist, so the word 'normal' always makes me cringe a little bit..."
It's a I-sentence, without accusation, also it says the truth (ie, you cringed) and it doesn't assume she's racist, just that she could sound like that to someone from a younger generation.
I'm not sure if it makes more sense to address her directly or to mention something about this to the leaders of the group to be addressed more broadly under the umbrella of racial sensitivity training.
Well, have you ever managed a non-profit?... It's simple, these people aren't paid, so the manager's top priority is to keep them happy. In other words, if she has been providing useful work to the organization for free for a long time, and you barge in and want to change everything, and then you appeal to authority even before you have proven your worth... well, any manager worthy of the name will show you the door.
If, as you suggest, you appeal to authority based on a judgment that might have been hasty then you're very much on your way to creating a toxic work environment, and that's the best way to lose people who work for free.
One single toxic individual can destroy a team, which is why it's management 101 to get rid of them ASAP. This sort is easy to identify, they're the ones that focus more on process (how we should do things) than on goals (which are what actually matter). So don't be the one to utter the words "racial sensitivity training". That's synonymous with "I'm gonna be a pain, please fire me."
If you want to suggest improvements in process (like "let's talk in a more inclusive manner" for example) you'll have to prove your worth first so they listen to you, and also explain how this isn't just your whimsical idea but something that will actually benefit the goal of the organization.
When you say "I'll bet that part of the reason she finds that the 'Blacks don't trust us' is because of the way she phrases things." this is a good example of it, except it's just a hunch right now, which you could study in more depth, find more arguments, perhaps ask the aforementioned people, etc.
Just talk to the old lady, framing it as a misunderstanding stemming from different ways of saying things for different generations. You'll be fine.