OK, since this is a big question which has been addressed before, I will address it part by part.
in your time in Japan, how prominent has anime been in it?
Far more prominent than in my native Austria.
Tremendously less prominent than the internet would like you to believe.
Imagine it, umm, around the level of the prominence of English domestic media in England? (I only use that example because I have been to Great Britain before, and Austria has next to no domestic media)
I ask because I know that people do often assume Japan to be like an anime Disney World
No. No, no, no. There’s theme parks for that, yeah.
Outside those theme parks, tho? No.
You can search events where you can geek off with others, but outside of those, you’d better keep your geekery to yourself the same way you would keep it to yourself in your own country.
and a safe haven for otakus, which is one of the worst stereotypes about the country and its people.
Again, no. There *are* places where you can totally geek out and just be happy to have your hobby, like the Otaku shopping streets in Akihabara or Ikebukoro. When there is an anime event at a cinema or an event space, you can geek out there, too. Most department stores have a fairly spacious sub-department for merchandise and DVDs/games/CDs/Manga, etc. But those are usually in the back of the store, where they don’t bug anyone not interested. And Arcades are plenty, but usually in sidestreets or underground. Again: Easily accessible for the target group, out of the way of people who could be bothered by them.
Outside of these spaces, you see anime sometimes in adds and on billboards and it’s kinda nice because of how it gives things color. But generally the rule is:
"There is plenty of stuff for you geeks to enjoy. Where it doesn’t bother anybody else."
So Otaku can casual fans can kinda keep to themselves, enjoy their silly little hobbies, be happy for the tons of street passes they get on the train, because it proves they are not alone, even out in the “Normal people’s world”, and whenever they need a break from the normal world, they just hush into one of their many, many little hidden-but-easy-to-track-down refuges to geek out properly.
Even Anime itself is mostly “hidden” in the midnight programming slots, where it doesn’t bother anyone (especially not the censors).
It’s actually kinda really nice.
As someone who’s been to Japan, I want to know how you’ve seen how that stereotype is right or wrong.
Japan is not Otaku Heaven. Get that out of your head. If you display your geekery openly here, you will have people avoid you like a pleague, because they will be convinced of all the negative connotations people associate with geeks not only in Japan, but almost *anywhere* in the world.
If you are happy with a myriad of tiny geek-hideouts that you can go to to just fangirl your soul out, I recommend coming to Tokyo, though. Don’t be surprised if people look weirdly at you because you’re a foreigner. Or a girl in Akihabara. They tend to do that. I recommend getting a fellow (japanese) geek-friend and going geek-spot-hunting with them. That’s *huge* fun.