Not exactly inaccurate, but not exactly to the point either. The point of the story is that vegetables by themselves would be considered an ascetic meal. 'Low and disdained' and 'paltry' are hyperbole to accentuate the point.

In reality vegetables were an important part of the ancient Greek diet, though not as important as in modern western cuisine.

The main component of the ancient Greek diet was grain, particularly barley and wheat (durum and aestivum), though also millet, rye, and oats.

The secondary components fall into three groups: (1) three four important legumes, namely chickpeas, lentils, broad beans, and peas; (2) meat and fish, typically as flavourings or sauces to accompany the main ingredients; (3) olive oil and wine.

But there were plenty of tertiary foodstuffs alongside these, including fruits, nuts, seeds, eggs, cheese, honey (usually as an ingredient in baked goods), and yes, vegetables.

Peter Garnsey's Food and society in classical antiquity (1999) doesn't have a dedicated discussion of these specifically, though he does mention vegetables frequently. There's a brief but more focused discussion in Victoria Tsoukala's PhD dissertation on The social context of food preparation and consumption ... (diss. Bryn Mawr, 2004), pp. 29-30, which suggests that the most common ones were celery, leek, beet, onion, cress, lettuce, and garlic. I'm a little uneasy that she relies on Andrew Dalby, but these do seem to be backed up by primary sources. I'm also aware that cabbage growing techniques are discussed at some length in Theophrastus and Pliny the Elder; and there were other cheaper or inferior options, e.g. cannabis foliage could be used as an inferior replacement for cabbage.

Some striking absences from the list of vegetables used would be carrot, artichoke, and most of the modern varieties of Brassica. When I said 'cabbage' above, that didn't mean the modern cabbage, which wasn't developed until the Roman era. It may have meant something closer to kale. Most of the modern varieties of Brassica seem to have become differentiated from one another in ancient Italy rather than in Greece.

The idea of a vegetable-centred diet as ascetic can be seen in some ancient Christian documents. The Life of Hilarion, dating to the 390s CE, describes Hilarion's ascetic diet:

... from the time [Hilarion] was twenty-one until he was twenty-six, for three years he ate half a pint of lentils soaked in cold water and for the other three years he ate dry bread with salt and water.

Then from the time he was twenty-seven until he was thirty-one, he sustained himself with wild herbs and uncooked roots of certain shrubs.

From the time he was thirty-one until he was thirty-five his food consisted of six ounces of barley bread and lightly cooked vegetables without any oil.

But when he sensed that his eyes were clouding over and that his whole body was contracting under the influence of impetigo and some kind of rough skin disease, he added oil to the diet just mentioned, and until the sixty-third year of his life he continued at this level of abstinence, tasting nothing else, neither fruit nor beans nor anything else.

(Quoted from Jason König's translation, in Saints and symposiasts, 2012, p. 342; my paragraphing.)

So the upshot is that I'd read the Diogenes story as an indication of Diogenes' asceticism, rather than as a marker of social class.