And in all this it's of course very confusing because leap seconds don't stop the passage of time over here in consensus reality, so surely this is meaningless and time counts up at one second per second all the time?
The key is that there's a difference between time elapsing, which it continues to do at one second per second all the time (let's not involve relativity please, my head hurts enough as it is), and the timestamps you strike according to a particular time standard.
If you strike a timestamp in UTC and TAI, wait 60 pulses of a 1 pulse/second atomic clock, and strike another pair of timestamps, they will both agree that 60 seconds have elapsed. What they won't agree on is what those instants in time are called. The TAI reference might say the start/end timestamp are 00:00:00 to 00:01:00, whereas the UTC reference might call those same instants in time 00:00:00 and 00:00:60, because leap second. Both time standards contain a timestamp 00:01:00, but they describe instants in time that are 1s apart.
When an API says it's returning the number of seconds elapsed since 1970-01-01 00:00:00 UTC, it could be saying two things: "an atomic clock that beeps once per second has beeped N times since 1970-01-01 00:00:00 UTC" (which would be the number of elapsed TAI seconds), or it could be saying "if you start at 1970-01-01 00:00:00 in the UTC time scale, and you increment the timestamp by 1 second N times, you will get the current time."
In the second definition, the key is that according to the UTC time scale, some minutes contain 61 seconds.
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