Update internship form data schema.

Changes the path to some data, renamed them or change their type.

To work with the backend, I just need to add the new `public` property

Also:

- make sure the user is register to view that partial or redirect him.
- remove prototype methods.
- switch to using currentUserApi.update to save the internship data.

I haven't check the admin controller yet, but it should also probably use the User services instead of create new endpoints.