I believe it is important to speak to kids on their level without dumbing down the vocabulary. If they need to know what something is called, then they should be told what that thing is. Whether it's a body part, an injury, an illness, or a bodily function.
I think it's equally important to teach them the proper words.
Males do not have pee-pees. They have penises and testicles/scrotum. (I'd opt for testes, to start, as it's easier to say).
Females do not have coochies. They have vulva. A vagina is not an obvious external feature, so the word may not need to come up very early. However, its wide misuse may cause it to be brought up.
Everyone has genitals/genitalia. I believe calling them private parts is appropriate, but not if that's solely how they're called. A penis is a private part, as is a vulva. This leads into explaining why they're private, when they should be seen, and when they should be talked about.
I don't believe in tummy aches. Is their stomach bothering them, or their gut? There's a big difference in preparation for about to throw up versus about to diarrhea.
There aren't boo-boos. You get hurt, a cut, a scratch, a scrape, a bruise, etc.
If you allow your child to have a stronger vocabulary then you can communicate with them more easily. You also encourage their observational skills.
While in many cultures it's not acceptable to speak about genitalia in public, it's also the case that children aren't always good about following social convention. I don't believe a child asking a question that others feel is inappropriate is reason to be embarrassed.
In fact, I don't believe it's right for others to make you feel embarrassed about your child exhibiting child-like qualities. Children will lose those qualities soon enough without others trying to speed up the process.
The correct response to a question you feel is embarrassing is to answer the question correctly, as if they asked you why the sky is blue. It's okay for that answer to be, "I will tell you when we get home if you ask me then." or "I don't know." Being outwardly perturbed sends the message to the child that they can't ask you about things, or teaches them they can get a fun reaction out of you by asking certain things. Calmly, matter-of-factly answering their question validates the child, encourages inquisitivity, and demonstrates to the surrounding public the appropriate way to handle sensitive topics in public.