Cittasanto: Could you give us some more examples of names of Dhammapada verses? The one you mentioned is the only one I can think of that might have a name. The Dhp commentary story to that set of verses is named "ānandattherapañhavatthu." Those verses also appear in DN14. They also appear in the Vinaya. In the commentary to the Vinaya (pārājikakaṇḍa-aṭṭhakathā, verañjakaṇḍavaṇṇanā, vinayapaññattiyācanakathāvaṇṇanā) they seem to be given the name ovādapātimokkhaṁ. The only other place I can find "ovādapātimokkha" is in the Udana commentary to the Uposathasutta. So it doesn't seem to be connected with the Dhp at all.
My experience with Sri Lankan monks is that the verses are referred to by the first words of the first line, not any sort of names.
If you don't get an answer from this forum, this seems like something Bhante Anandajoti would know the answer to.