If you've already seen a lot of suggestions about what constitutes "business casual," but you're confused because it seems to cover a wide range, here's my advice: Go with the LEAST casual option. If some of the things you're seeing suggest a tie, and the others don't, wear a tie.
Why? It's never a bad idea to be the best dressed person in the group. If an organization has a room full of interns and they're thinking about offering one or two of them a permanent position, they're going to pay attention to the most professional ones first. And part of what makes you seem professional is your choice of outfit.
Also, it's much easier to be overdressed and then dress down from there than it is to make your first impression as "too casual" and then convince an employer that you could dress up more if they wanted you to. First impressions are important.
The number 1 rule, though, is "know your audience." Look at how the permanent employees are dressed at various levels. Nobody is expecting you to turn up to your internship dressed the same way the corporate executives are dressed. But watching them will give you a good idea of the organizational culture. If you're trying to turn this internship into something more permanent, it's important that you show them how perfect a fit you are for them. And looking the part is a big step in the right direction.
So dress UP, knowing you can always dress down later.