Technically, “the Holy Spirit” isn’t a proper noun. That’s because it contains an adjective also. Personally, I wouldn’t capitalize any of those terms. The head of the Catholic church only really when talking about it as a title in front of the assumed name, Pope Leo. Somebody’s style guide may have them capitalize all nouns with religious notoriety. That’s a style choice then rather than grammatical necessity.

Also, grammar is the attempt at formulating rules about how people talk. Meaning the talking came first and then somebody tried to codify it. That means there is virtually no grammar rule that doesn’t come with at least one exception. Learning when to use an article or not, when it should be the indefinite or the definite one, is killing students who come from languages that don’t use articles at all. And in some cases also those who speak other article-infested languages. That’s because it doesn’t make any sense and you had to grow up being corrected by other speakers to get this mostly right. It is convention more than clean cut rules.