NavList:

First off, with regards for terminology for the difference in ΔT between actual values and predicted values in almanacs, in the sections discussing solar eclipses in the Astronomical Almanac and Astronomical Phenomena the symbol "δT" is used.

Secondly, with regards to subjects like ΔT and leap seconds in UTC, probably the best source of info is:

https://www.ucolick.org/~sla/leapsecs/

From here down, I'm just going to try to unpack and summarize timescales and ΔT in terms of celestial navigation.

ΔT = TT - UT1

UT1 pretends to be based on a "mean soloar day" but is actually GHA Aries in disguise.  The Nautical Almanac has GHA Aries increase by 15°02.46' per hour UT1 because that is the definition of an hour in UT1.  (The sun has v-corrections that the Nautical Almanac hides in rounding errors.)

TT is the symbol for "Terrestrial Time."  TT is the scale we try to emulate with atomic clocks.

Everything in the sky has declinations and SHA's that are measured from Aries (the almanacs list GHA of solar system bodies just to save you from doing the math yourself).  All of these motions about Aries have nothing to do with the earth's rotation (i.e. nothing to do with GHA Aries).    So astronomers' predictions of their motions with respect to Aries and each other is based on TT, not UT1.

When a navigator takes a lunar distance, they are measuring TT directly. The almanac's assumed ΔT value will take the navigator from TT to an assumed GHA Aries, which is then compared to LHA Aries to determine longitude.

For the navigator, GHA Aries races along at more than 15° per hour, but it takes about 30 hours for the "speedy" moon's SHA to change by the same amount.  Note that the Nautical Almanac's "Increments and Corrections" tables only tabulate d- and v-corrections for the nearest minute.  An error of several seconds in TT, by itself, is almost imperceptable with a sextant.

Although TT and UT1 are physically independent of each other, they "tick" at very nearly the same rate.  It's so close that it took the development of quartz chronometers to verify the difference.  Comparing spring-driven chronometers to measurements of LHA Aries at a known longitude before embarking on weeks-long voyages was clearly accurate enough for the ignorant navigators of the 18th and 19th Centuries.

Counting from near the beginning of the year 1900, ΔT has grown to about 69.4 s. 

ΔT is made of several parts:

(TT - TAI) + ΔAT + (UT1 - UTC)

TAI is International Atomic Time.  (TT - TAI) is a fixed value of 32.184 s (0.000 3275 days exactly).  When the atomic clocks were first started, they were set to UT2 and the fixed value represents the difference between TT and UT2 at that particular moment.

ΔAT, the difference in atomic time scales, is (TAI - UTC).  It's the effectively the number of leap seconds that have been inserted since TAI was started.

UT1 - UTC is what's approximated by the DUT1 coding in international radio time signals.

"UT1 - UTC" has to be specified because there are enough different versions of "UT" for "ΔUT" to be vague.

UT0 is determined by a single observer determining LHA Aries from a known longitude and an assumed latitude.  The latitude has to be assumed because the earth's axis moves slightly with respect to the surface ("polar variation"), causing latitudes to wiggle on the scale of 0.005'.  A single observer can essentially produce a single line of position for their elevated pole.

UT1 is determined after enough observers coordinate for a solid fix for the pole at the time.  UT0 was a necessary step on the way to UT1 when it took days or weeks for data to be collected through the mail, but today's global, real-time communications allows UT0 to be invisibly "baked into" the process (analogous to getting a fix from 4 satellites simultaneously).  The poles vary around 0.005', so the maximum difference between UT0 and UT1 is that angle converted to time: about 0.02 s.

UT2 is an attempt to smooth out small, cyclical "wiggles" in the earth's rotation.  Part of the difference in (TT - UT1) is predictable, on the scale of +/- 0.02 s over the course of a year.  UT2 was supposed to be the "end result" sent out by international radio broadcasters (UTC started as a coordinated effort between the US and UK to determine UT2), but it was a solution in search of a problem.  People who aren't trying to find GHA Aries don't care about precisions finer than 1 s, and those who are trying to find GHA Aries want to find it "now" not "now plus a smoothing factor."