What Spock describes in Amok Time does sound alot like a mind meld, and may effectively have been a version or variation of that.
The key thing to understand here though is that mind melds were no longer a forbidden practice during the time period that The Original Series was set in (the 23rd century). They were a forbidden practice during the time period the Enterprise series was set in (the 22nd century). At that time, most of Vulcan society had drifted away from the teachings of the fourth century Vulcan philosopher, Surak, who was an active mind melder. The minority of Vulcans who still practiced his teachings, known as Syrrannites, were frowned upon by mainstream Vulcan society, regarded as dissidents at best, and terrorists at worst, even though they were pacifists.
This all changed in the Enterprise episode, Kir'Shara, when Jonathan Archer -- who at the time was carrying the katra of Surak within his mind, as the result of a mind meld with Syrran -- discovered an ancient Vulcan artifact known as the Kir'Shara, and brought it to the Vulcan High Command. This effectively legitimised the Syrrannite movement in the eyes of mainstream Vulcan society, and allowed them and their newly-instated leader, T'Pau (who also appeared as a much older woman in Amok Time), to become highly influential in the shaping of Vulcan politics and culture from that point forward.
As an addendum, it was possible for Vulcans to form a psychic bond with one another, or with humans (and possibly other humanoids), without undergoing a conventional mind meld. This came up in the Enterprise episode, Bound, when T'Pol explained to Trip that a psychic bond had formed between them as a result of them having had sexual relations some time earlier, in the episode, Harbinger.
T'POL: There's a long-held belief that when a Vulcan mates, there's a shared psychic bond.
TRIP: We didn't mate.
TRIP: Okay, what do you mean by psychic bond?
T'POL: It's difficult to explain. Feelings, thoughts, even images can be shared.
This would appear to be distinct from a conventional mind meld on at least two counts:
This bond apparently formed by accident, whereas mind melds (so far as I've seen) have always been portrayed as an intentional act, at least on the part of the Vulcan initiating the process.
Mind melds typically involve an extensive merging of two minds, to the point where the individuals involved know each other's thoughts and feelings while the meld is taking place, and retain an impression of the other person's mind for some time afterwards. Whereas Trip and T'Pol did not appear to retain an impression of one another's minds, thoughts or feelings. The primary manifestation of their psychic bond was a series of shared "daydreams", where they could see and speak to one another within their minds, for short periods.
This instance of psychic bonding was one of several examples listed on the Memory Beta page for Vulcan bonding, along with the type of bonding said to take place within the koon-ut-la ceremony. The main thing that would seem to unite these examples, and set them apart from the mind melds we typically see on-screen, is that these psychic bonds are specific to those have mated, or who are intended to marry in future. It seems very possible, if not probable though, that these bonds are effectively another variation of a mind meld.