Well the absence of pointy-eared people and the fact that the episode was apparently set in 2020 implied that.
Honestly, I suspect that TOS's writers may have originally intended Spock's Vulcan to be the cis-Mercurian one. After all, Roddenberry suggested that Spock was "possibly half-Martian" in his original proposal, so he started out thinking of Spock as being from a Solar planet; and it wasn't until "Amok Time" that Vulcan was portrayed as being in its own distinct system (since the ship was described as being on course for Vulcan rather than for Sol System).
Though I am curious as to when people stopped thinking that there was a planet between the Sun & Mercury. I remember reading James Blish's adaptation of 'Tomorrow is Yesterday'. I believe his version was based on an early draft of the script and when Captain Christopher first meets Spock he mentions that Vulcan until Spock disabuses him about its existence.
Urbain LeVerrier proposed the existence of Vulcan in 1859 to explain some observed anomalies in Mercury's orbit, on the principle that some unknown planet was tugging on it gravitationally. But in 1915, Einstein's equations of General Relativity explained the discrepancies exactly as the result of spacetime warping under the Sun's gravity, so the existence of Vulcan was disproven. Still, the idea had caught on in pop culture, and so it lived on in science fiction for a few more decades.
https://en.wikipedia.org/wiki/Vulcan_(hypothetical_planet)