nobody outside the left uses this example because it emphasizes that latter point, but it's really the best way to understand the difference. it's not exactly marxist, but it's along those lines (and let's not create strawmen about stalinism, which was antithetical to any kind of freedom whatsoever). consider the issue of employment. if you are a member of the working class - that is, if you must survive by selling your labour - then you are said to have negative freedom if you live in a free labour market. that is, you are free from external restraints (ideally; not if there are race or gender barriers) to pursue the type of employment you desire, after you pursue the proper education. but, because you are a member of the proletariat, you are unfree in deciding whether to work or not. that is, societies centred around markets lack positive freedom in the most basic sense of how we choose to spend our time, unless we are exceedingly lucky or exceedingly talented. the conclusion is that negative liberty is the kind of "liberty" we assign to the lower classes, while reserving true positive liberty for the elite.
once this class division is firmly cemented, this argument about taxation being necessary for or even somehow connected to positive freedom gets trotted out - as though the only way for one worker to be more free is to take something away from another, because the elite is shielded from redistribution (both of resources and labour). the only reason you need to talk about taxation and redistribution is because the system is designed to make positive freedom almost impossible in the first place. even the examples people use - of a well-paid profession having positive freedom because they have resources - are mostly nonsense, because that supposed positive freedom is a consequence of it's non-existence. it reduces positive freedom to a type of feudalism; it presents it as an award for the paying off of one's debts. it is simply a class division on what freedom can be conceived as.
the contradiction arises simply from the class division; they're only contradictory ideas in a society divided by class. the division merely serves to uphold the capitalist status quo. so, of course libertarians and (right-leaning liberals) see the issue through this lens.
however, if we were to abolish property and currency (and hence slavery) then we would see that there is no real contradiction - that positive freedom is required for true negative freedom and vice versa, and both ideas, to be actualized fully, require that abolition.
to put it another way: we could define a free society as a society where negative and positive liberty are not contradictory but cumulative to a coherent, unified concept of liberty.