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This is a simple question about terminology and provenance.

I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture.

I've encountered Polignac's conjecture generalizing the twin prime conjecture by replacing pairs $(p,p+2)$ with $(p,p+k)$ for any even $k$. The even more general Hardy-Littlewood conjecture actually predicts the density of any prime constellation not ruled about by residue considerations.

Does the conjecture weaker than Hardy-Littlewood merely predicting the infinite occurrence of such all such constellations enjoy a name unto itself and/or a known first appearance in the literature?

Are there other (canonical) conjectures that belong to this family, either over the integers, or other rings?

This is a simple question about terminology and provenance.

I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture.

I've encountered Polignac's conjecture generalizing the twin prime conjecture by replacing pairs $(p,p+2)$ with $(p,p+k)$ for any even $k$. The even more general Hardy-Littlewood conjecture actually predicts the density of any prime constellation not ruled about by residue considerations.

Does the conjecture weaker than Hardy-Littlewood merely predicting the infinite occurrence of such all such constellations enjoy a name unto itself and/or a known first appearance in the literature?

I should also mention that I've encountered the Bunyakovsky conjecture and Schinzel's hypothesis H which generalizes both Bunyakovsky and the (as yet for me) unnamed conjecture of the previous paragraph.

Are there other (canonical) conjectures that belong to this family, either over the integers, or other rings?