Instead it's a shrine. It may be situated inside a religious building, but it's not built as part of it.
This is the Ascension edicule on the Mount of Olives:
photo by en:user:adriatikus
And this is the Jesus Tomb edicule inside the Basilica of the Holy Sepulchre:
photo by Joe Goldberg from Seattle.
Now, this is all very well. These things exist, and so they need a name.
But a word describing the contempt of the uneducated for the highly schooled would be ever so much more useful.
Word Not To Use Today: edicule. This word comes from the Latin aecula. Aedēs means temple or shrine, and is related to the Greek aithēr, bright upper sky. The -cule bit is a Latin way of indicating something small.