Radical is a loose word. It can mean both "part"/"component" or "part used for classification".
In dictionaries or kanji-learning ressources, it often means the latter. But it's the only place where it's used really strictly in that sense. If you go elsewhere (especially broader linguistics), you will find expression like "semantic radical" or "phonetic radical", etc.
Personally, when I want to have a serious discussion about kanji, I say "key radical" (most people don't know/use that expression but at least it make it clarify what I am talking about). On renshuu tho, I just go with whatever definition people used. When I'm talking to japanese too. Because most Japanese don't differentiate "radical" from "part" (ironically...).
As for the initial question:
"Why does the kanji 怒る (to get angry) use the woman radical (女)?"
No one knows =p
Etymologyst refuse to answers because they can't scientifically prove it. Like they can't read the thoughts of people that lived a thousand years ago. So they just don't answer.
On the other hand, finding some phonetic relations is easier. Like
女 (じょ, it's close to "djo" phonetically) -> 奴 (ド, do) -> 怒る (ド, do)
But beyond that, they refuse to make any comment. It's not that 奴 don't have any meaning. It just that there is not valid method to prove it.
For example, one interpretation of 奴 is 女 (woman) + 又 (手, hand, it's just another way to write hand). So you have the image of a woman using her hand, so a "female servant". And after that, by extension, just "servant".
So, when you look at 怒, you may think of servant getting agitated, thinking of revolting or something like that. Hence the current meaning.
BUT all of them are hypothesis. There are not proven.
Another interpretation is 奴 though as "labor". In that interpretation, 奴 convey a sense of "pressure" (as "force to do some labor"). And 怒 is interpreted as something like "pressure on the heart" or "pressured emotion" => anger.
That kind of interpretation is done by looking at multiple characters and try to see if there is a common "semantic field/pattern". In this case: there is also 努 (serving but also effort, tension, etc).
You can remember those interpretation as a sort of menmonic but don't think it's the truth. Because even the linguists working on that kind of things dare to be so confident.
PS: Also, regarding your initial question, 女 probably didn't hold any meaning anymore in 怒る even those "free" interpretation. Even if 奴 did probably carry the woman meaning initialy, it was probably already lost later long before it was use in 怒る. But again, that's too, is an hypothesis =p (Fun to think about but don't go anywhere saying it's the truth, because that's incorrect)
EDIT: I'm a bit too biased here. When I said "it's the only place where it's used really strictly in that sense", it's not true. But at the same time, it's still far less clean than just "Radical=Indexing component". Linguists disagree with each others, Etymologists disagree with each others, dictionary makers too (even if most will still stick to that definition), etc.
