Around this period you often had the paupers marked because the incumbent had to collect fees and pass on a proportion, so the marking was to keep tabs. "On request" might be a specific indication that the fee was paid.
But at this period you have the dictates of an individual bishop, its interpretation by the Archdeacon, and the vagaries of the clergyman to contend with.
I have an ancestor born at that period who joined the Navy. To prove he was of age to become a lieutenant, he had to produce a baptism certificate. The next candidate had affidavits from lots of people because the then incumbent had not seen fit to record private baptisms.
As OC says, you can't actually infer anything. And if they were a nonconformist family, they were probably sufficiently well off that there wasn't a high infant mortality to give the game away.
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The chestnuts cast their flambeaux
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