In home daycare providers do not pay your taxes for you. They pay their own taxes on the income that they earn. You pay your own taxes on the income that you earn and then use to pay the daycare provider. It is always the person who earns income that pays the taxes on the income that they earn.
Employers take out taxes as a convenience for you before paying you. It's called tax withholding. The employer then sends the money to the IRS. When it comes time to file you have already paid taxes and may not have to pay more but if you didn't have enough withheld you have to pay the difference.
So, if you contract out for daycare you pay the daycare provider but you do not withhold taxes for her because you are an individual person. If "you" are a company that contracts out to daycare it may be that the company would withhold taxes.
I think you're trying to put too much thinking into this or I misunderstand your question. Here's my simplified answer. If you, as a parent, hire a day care provider, in home or otherwise, you pay their fee. They pay their own taxes.
If you hire a nanny that comes into your home you may be required to withhold taxes. In that case you would contact the IRS to receive the appropriate forms. I'm not sure if you're required by law to withhold taxes from the wages of domestic workers. I do think it's an option.
Perhaps you're confused about the term in home daycare. This means that the daycare is done in a private home, not the home of the child, as opposed to an institution or business setting. I suspect that the two documents are focused on different situations. Perhaps one is for a nanny situation and one is for an in home, not the child's home, situation. I'd pick one category and not try to compare or figure it out using information on a different category. Perhaps both pamphlets use the term in home. One pamphlet is about in home but not in the child's home and the other pamphlet uses in-home meaning in the child's home.
If you hire a nanny or child care from an organization that places employees, then you may be considered an employer. When an office hires a temp employee they give the agency the money and the agency withholds taxes. It is very confusing.
It sounds like you want to place your child in another person's home for child care. Focus on just that. You are not an employer. You are a client. You do not pay or withhold taxes for the daycare provider. They have several clients and there would be no way that an accurate amount could be withheld.