Interesting thread. I do feel that perhaps the idea is a good one here.
My vote was for either. As touched on already, the prepositions that follow verbs in those phrasal verbs can make a difference in extent or degree, quality or quantity, causative linkings or non-causative linkings. Also, of course, context has influence on how things are taken as well.
As far as any given quantity of a fixed nature goes, to say that the fixed quantity is lacking is equivalent to saying that the given quantity is not. If there are to be four things understood as being a possible defined set, to say that set in which those four things should be the given quantity lacks, is to say that there are less than four--there are not four.
Within the context of a single sentence in which that clause can be seen as the independent portion of the sentence, the elipsed dependent section can be understood in the light of 'carry over'. Though the concept in the mind of the source may be focused on a lesser degree of quality--which stipulates a comparative function--the recipient can understand the state of quality from which the comparison had been likely intended to be not in that which had be expressly stated. To say that a given set A lacks a given set B,
can be seen to equate the essence of the statement that set A does not have set B.
Another point, if I may, could be made to show how context works. You are correct in your examples given at the bottom of you post, Maciamo san, and I will take one and apply it here:
A lack of money is a general statement, and so as you have paraphrased it, presents the idea of a comparatively lesser quantity. However, to say
I lack the money, as I think you know, presents the idea of not having a given quantity--even being unidentified. In other words, the given quantity to be possessed is not.
In my post on that thread, I think you'll notice that I clearly presented the idea of comparison for the later clause of that title. The first clause, as pointed out there (but perhaps not so clearly [I aplogize for that]) and here, stands, I firmly reason. I think it is nice that you have pointed out the matter here.