I'm still not making myself clear. Your lemma says that having having arbitrary joins suffices, you do not need distributivity in order to be closed. But didn't you give an example in comment #29 with joins which is not closed? Does your example not contradict the lemma?

This is not about John's definition of quantale as he assumes closedness, by the adjoint functor theorem for posets, as you point out, this will have the distributivity property. We are focussing on your lemma and your example of a non-closed monoidal poset.