All plums are Prunux x domestica, a variable group of plants of hybrid parentage involving Prunus spinosa (native European blackthorn or Sloe) and Prunus cerasifera var. divaricata (cherry plum, myrobalan, native to eastern Europe and central Asia). Within this already variable group, three main subgroups are noted: P. domestica ssp. domestical (plum); P. domestica ssp. interstitia (damsons, bullaces, mirabelles); and P. domestica ssp. italica (greengages).
Given that these are all closely related and can cross-pollinate freely, there are bound to be countless intermediate forms. So precise classification of any kind of un-named plum-like fruit is never going to be exact.
Bullace is the common term for a wild plum, and seems be used interchangeably with damson. My own interpretation of the difference is based on the difference in eating qualities between both, rather than size, shape etc. The unique quality of a damson is in the high levels of both tannin and acidity in the skin/flesh which renders the fresh fruit virtually inedible, but valuable for jam and winemaking, where high acidity and tannin levels are actually desirable qualities. Bullaces have a fairly bitter skin, but slightly sweeter, less acidic flesh but not to the degree that you'd actually ever prefer them to a desert plum.
What I call 'damsons' are those fruits which nearer to sloes in size, shape and flavour; 'bullaces' the type which are rounder, with a higher ratio of flesh to stone, and less tannic astringency.
The wild plums that infest the borders of my own garden vigourous, spiny things, with round, serrated leaves and round blue/purple bloomed fruit (photo 2). My in-laws have a range of wild plums in their orchard in south Warwickshire; theirs are smaller in height and leaf size, more oval dark blue/purple fruits and leaves; are less spiny and are mouth puckeringly tart (photo 1). These are referred to as 'damsons' locally. I'm not sure the distinctions are really that important.
Other thoughts on the matter:
Keeper's Nursery describes Small Black Bullace as "Very small round fruit. Blue-black skin with a purple bloom. Firm juicy green flesh. Clinging stone. Acid favour. Recommended for jams."
In comparison, Shropshire Prune, ' a classic damson' : "Small, oval fruit. Blue-black with a dense bloom. Strong, rich, astringent damson flavour. Considered to be the best flavoured damson.
'Compact tree with dense twiggy branches. Fair and regular cropper but never producing very heavy crops like the Farleigh damson."