Any gift to you is a non-taxable gift. I don't know what you're reading that is blurry on the topic. If you perform a service for money, that's income, but free will offerings for a person in trouble, whether made by individuals or the church, is a non-taxable gift.
It's the church's responsibility to comply with the tax laws on deductible donations, you don't need to concern yourself with that at all. However, since you seem interested, gifts to specific persons, even if they are needy, are never tax deductible. Gifts to the church can be deductible if the giver does not control where the gift is going. So if the church says, "please help the Smith family with their move", the givers can't deduct their gifts, whether they are made directly to you, or passed through the church bank account, and the church is not supposed to give them donation receipts. But if the church says "We want to set up an emergency fund for a family in need", and the decision of which family to bless is made by the pastor or the board of Deacons or Elders or whatever, then donations to the church emergency fund are probably tax deductible to the givers. But as I said, that's the church's responsibility, not yours.