Not a bad list and a good idea generator.
Two comments, in case anyone cares.
First, for people looking for an introduction to classical music, please understand that there is a vast gulf of difference between the 'greatest' music and the 'easiest to enjoy'. While this might be a good list of music to aspire to appreciate, it is not a good introduction to the several different genres included within it.
Second, and perhaps it is the iconoclastic contrarian in me yet again, but I shudder at the ill deserved inevitability of seeing Beethoven's 9th symphony at the top of every list. Yes, it is a good piece of music, and yes, the 'Ode to Joy' is a decent enough tune to walk away with at the end of listening to it, but the best piece of music ever written? The 'most popular' and 'best known' - definitely. But greatest???
Note the quotes around 'most popular' and 'best known'. I doubt many people can appreciate or enjoy the third movement, and I'll even wager that many people skip the first two movements, and impatiently wait for the introductory parts of the fourth movement to be over and done with too. Indeed, while it also comes near the top of awful lists like Classic FM's top 100, probably most of the people voting have never heard the first three movements and don't even know they exist!
I'll avoid a discussion of what makes a piece of music great/greater/greatest, and simply say that I think Beethoven himself wrote better music. Symphonically, the 3rd symphony, is a great rollicking ride. More abstractly, the imponderableness of his last piano sonata (if played well, and few people do) leaves me every time feeling I've had a briefest of glimpses, through an otherwise impenetrable mist, of a transcendental experience that brings me closer to God, and leaves me humbled (and puzzled).
And, because so many others have shared some glimpses into their own personal top 100 lists, may I thank the OP for mentioning Elgar at all, and commend his Enigma variations to all for consideration (see if you can guess the theme they are based on - the mystery, not due to be revealed for several decades yet, frustrates many of us and my money, little as it is, is on 'Twinkle Twinkle Little Star'.
Similarly, I see Richard Strauss' Der Rosenkavalier, but do not see his Four Last Songs, the last two of which again bring me somewhere close to that spiritual place (but via a very different and much more approachable path) that the Beethoven op 111 sonata also does.
Definitive versions? Well, in the case of Elgar, we are fortunate to have one or possibly two versions conducted by the composer himself available on disc. Not the best audio, but it is wonderful to hear the composer's own realization of his score. As for the Four Last Songs, there's one version that is so far ahead of all others as to allow for no debate - Berlin Radio SO, Szell and Schwarzkopf.