As has been noted "classical" in the context of psychedelics is a cultural invention and a somewhat loosely defined term. But in my view classical psychedelics are those that produce their psychedelic effects primarily via serotonergic mechanisms, specifically (but not exclusively) action on the 5-HT2x family of receptors, primarily 5-HT2a. In that sense I would consider tryptamines to be the most "classical" family of psychedelics, followed by lysergamides which contain tryptamine groups and exhibit similar binding profiles, followed by phenthylamines which tend to have more complex receptor binding profiles and a wider array of effects. Dilution of effect profiles more suitably described by other effect-oriented class descriptors, such as stimulants, for example, in the case of MDMA, IMO makes a psychedelic less "classical", for example I would not consider MDMA to be a classical psychedelic even though it might have psychedelic effects, nor, in fact, would I consider most of the phenethylamines to really belong to the "classical" descriptor in the strictest sense.
Of course there is also a historical context to the usage of this term, which in my view narrows down the field somewhat from the vast array of substituted tryptamines and more recently discovered or just very rarely synthesised compounds with very little known about them to, probably, the first few discoveries in any given chemical grouping that enjoyed some period of widespread use. In that sense, LSD as the first lysergamide definitely meets all the criteria to be called a "classical" psychedelic, as does psilocybin, the first isolated tryptamine which historically is also surely the most widely used given it's occurrence in nature - I'm going to extend this description to psilocin too for reasons that probably don't really need to be explained. The latter point about use in nature obviously applies to mescaline too, and of the phenethylamines, in deference to Alexander Shulgin, I would allow that 2C-B could be considered to be classical as well as the first and probably most widely used substance of this particular class.
So in my view in the strictest sense the classical psychedelics would include LSD, Psilocybin Mushrooms, synthetic psilocybin or psilocin, mescaline, and 2C-B.
Personally I would say that definition should be broadened a little to include basically all known lysergamides given how similar they are to each other - unless one is discovered which is radically different somehow - most, maybe all, of the 4-substituted tryptamines - maybe all tryptamines, in fact - and most of the 2C-x class. Maybe most mescaline analogues too.
One final thing I would add is that in a cultural context "classical psychedelic" is typically understood, I think, to refer to a substance which, physiologically, is very safe - and this definition does not apply to a larger proportion of the phenthylamines which is why I am a lot more hesitant to include them under the "classical" umbrella. For example Bromo-DragonFLY, 2C-T-x compounds, the DOx family, are significantly more dangerous and therefore I would exclude them from the "classical" descriptor.
Dissociatives of course are not classical at all and although some of them can produce psychedelic effects I do not believe them to be real psychedelics and have argued that point many times on this board, so I'll just reiterate it here in case anyone was wondering.
Obviously there are probably some edge cases I haven't considered - but again, it's a hazy term, that's just unavoidable. But, that's my interpretation of the kind of substances that "classical psychedelic" refers to, and LSD fits very very solidly into it.