If anything, I'd interpret the "Classical" moniker to mean the defenses that play e6. as in the Lasker variation, The Tartakower Variation or my favorite the "Orthodox Variation" The Orthodox is solid as a rock, never been refuted and is the where many of the "newer" Queen's Gambit Defense got their origins. At one time, the Orthodox was the primary defense. I continue to play it. Like the Petroff... the Orthodox Var is a defense with such defensive capabilities, White has to play extremely exact, to get, and to keep any advantage... hence resulting in the high % of draws in it's heyday... Now the theory (which is as complex as it gets) is unknown to the masses (Bonus!)
The Orthodox hype aside... Both Bg5 and Bf4 are part of the Queens Gambit (so is playing e3 and and temporarily shutting in the DSB)
Bf4 var is known in the United States as the Horowitz Attack, named after ISRAEL ALBERT HOROWITZ, he was one of (if not the best) American players in the 1930s. He authored a lot of great chess books. It's also known as the Blackburne Attack in Europe, Named after Joseph Blackburne, a famous English Chess player Pre WWI
The Bf4 var as far as tactics goes is IS RIGHT UP THERE ON PAR with the Sacilian Nadjorf, the Semi Slav Botvinik var.. in fact I've read some expert think it might surpass them as far as complexity goes. Whatever the case, you are entering into a pure tactical matrix if you play the Horowitz Attack!
Variations where White plays Bg5 are positional in nature, both sides tend to fight over small advantages, i.e. who can stiff who with an extra move in the opening (i.e. the fight for tempo) Some variations like the Tartakower have in one of it's main lines hanging pawns at d5 & c5... Yet..... in variation where e6 is played first (excluding the Slav and Semi Slav varaitions) about 50% of the time the resulting pawn position will be the famed "Carlsbad" pawn formation where White will launch a minority attack and Black to off set this, will launch an attack on White's king.
Good stuff all
ECO calls D37 the Classical var. , giving typical line 1 d4 d5 2 c4 e6 3 Nc3 Nf6 4 Nf3 Be7 5 Bf4 . Yet a chess.com lesson titled QGD classical variation, by GM Kaidanov, shows 4 Bg5 . Running his stem through 4 Bg5 in 365chess Opening Explorer yields D50, and could lead to a variety of lines D50-D69 including Lasker, Tartakower & Orthodox but not "Classical".
van der Sterren in FCO does not use term "Classical". He calls above, through Bg5, the Capablanca line. Ntirlis in Playing 1 d4 d5, A Classical Repertoire simply lumps lines into QGD Bg5, QGD Bf4 and Exchange. He gives no other names for these lines and never uses ECO codes.
chesspub does not use "classical". It just discusses 5 Bf4 in same section with Tarrasch, semi-Tarrasch & Exchange var.
Studying openings has been compared to having dental surgery. Hard enough without agreement on names. When I use a word it means just what I want it to mean... Humpty Dumpty.