IRS Pub 517 partially addresses the question posted here:
"Teachers or administrators. If you are a minister employed as a teacher or administrator by a church school, college, or university, you are performing ministerial services for purposes of the housing exclusion. However, if you perform services as a teacher or administrator on the faculty of a non-church college [including non-church schools and universities], you cannot exclude from your income a housing allowance or the value of a home that is provided to you."
The principles communicated in Publication 517 apply to a Christian school teacher’s classification as a minister. In the case of the music minister, it appears that he meets the definition of a minister for purposes of his employment at the Christian school. Accordingly, he continues to be eligible for his exemption from SE tax. Further, as a qualified minister, he is not subject to FICA tax withholding.