'Earth-based' adjective counterpart to 'Lunar' or 'Solar'
Solution 1:
Terrestrial:
- relating to the earth
(Cambridge Dictionary)
Solution 2:
Earth is "Terra". The corresponding adjective would be "Terran".
https://en.wikipedia.org/wiki/Terra
"Terran" can be used for a person or life-form from Earth, to contrast them from those from other planets or places. Taking the intention from the title of the question, an earth-based life-form (as opposed to one from Mars) would be "terran".
"Terrestrial" is more used to distinguish things that are on the ground to distinguish them from things that are in the water ("marine") or airborne. Hence "terrestrial snakes", "terrestrial travel".
I would say "earth-based" in most cases would be "terran", but that in the case of eclipse, "terrestrial" is more conventional.
Solution 3:
As user110518's answer states, terrestrial is the required adjective. However, attacking the "eclipse" angle...
A Solar eclipse is when we cannot see the Sun (because the Moon is in the way):
/ \ ____ /
/ \ / \ |
| Earth | ==> | Moon | | Sun
\ / \____/ |
\ / \
A Lunar eclipse is when we cannot "see" the Moon (because the earth [mostly] blocks the Sun's light from reaching it):
____ / \ /
/ \ / \ |
| Moon | <== | Earth | | Sun
\____/ \ / |
\ / \
From these, we can take it that an X eclipse is when we cannot (fully) see "X" because something (unspecified) is in the way.
Thus, someone on the Moon as the Earth passes between it and the Sun (the arrangement we call a lunar eclipse) would experience a Solar eclipse (because the Sun is blocked):
____ / \ /
/ \ / \ |
| Moon | ==> | Earth | | Sun
\____/ \ / |
\ / \
The reference that Sven Yargs pointed out in a comment, The Observer's Guide to Planetary Motion, 2014 by Dominic Ford confirms this [emphasis mine]:
As seen from the Moon, the Earth's disk appears to pass in front of the Sun at a lunar eclipse. [...] It is rather curious to think that what we on Earth call a lunar eclipse might reasonably be called a solar eclipse by an inhabitant of the Moon.
He (Dominic Ford) then goes on to introduce the term terrestrial eclipse, but I think he does so incorrectly [emphasis again mine]:
Likewise, what we call a solar eclipse on Earth might reasonably be called a terrestrial eclipse on the Moon. If our hypothetical lunar inhabitant were to see a partial solar eclipse—the Earth's disk partially covering the Sun—then the part of the Moon's surface beneath his feat would lie within the Earth's penumbra. However, if he were to see a total solar eclipse—the Earth entirely covering the Sun's disk—then he would be standing within the Earth's umbra.
The problem is that although he starts talking about a solar eclipse from the Earth's perspective (where the Moon is in the middle; 1st diagram), the remainder of the paragraph describes a solar eclipse from the Moon's perspective (when the Earth is in the middle; 3rd diagram). This arrangement, we have already agreed, is a solar eclipse because the view of the Sun (from the Moon) is blocked.
One plausible definition of a terrestrial eclipse would use the same arrangement that we (on Earth) call a solar eclipse:
/ \ ____ /
/ \ / \ |
| Earth | <== | Moon | | Sun
\ / \____/ |
\ / \
In the same way as a lunar eclipse is where light falling on the Moon is (partially) blocked by the Earth we're standing on; here, a terrestrial eclipse would be where someone on the Moon, looking at the Earth, sees [a small] part of it obscured by the Moon they are standing on.
However, just as a full solar eclipse (where the Moon completely blocks the Sun) can only be seen from a small area of the Earth; here, only a small area of the earth would be dimmed by the Moon: it is, perhaps, only technically an eclipse.
The other definition of a terrestrial eclipse – and one only very recently available – would be where the observer's view of the Earth is blocked by something else. This would never have been visible from the Moon (there's nothing large-enough between the Moon and the Earth to get in the way): it is only recently1 that space-probes have offered us such a view (although this is still only a partial terrestrial eclipse:
/ \ ____ +--------+
/ \ / \ |DSCOVR |
| Earth | | Moon | <== |L1 point|
\ / \____/ +--------+
\ /
1 The most recent, and (presumably) best images of this are available on NASA's From a Million Miles Away, Moon Crossing Face of Earth page. There, it also notes that NASA's Deep Impact spacecraft captured a similar view in 2008, but only from a distance of 31 million miles.
Solution 4:
It's always 'terrestrial' with reference to an eclipse. For what it's worth, 'earthian' isn't the adjectival form of 'earth', though. The actual words are
Earthly
(attest. from OE) Of or relating to earth or Earth, variously
Restricted to Earth; hence non-spiritual, material; hence worldly, base, coarse.
On Earth; hence (obs.) on or in the ground or (in quest. & neg., esp. before 'no' or 'any') in any possible way.
(arch.) Made of earth.
Like earth; hence (arch.) ready for a tomb, pale and lifeless.
(chiefly SF) From Earth or resembling sth from Earth.
(n., in plural) Earthly things or people.
(n., UK colloq.) An earthly chance.
Earthling
(attest. from OE) A being of the earth, variously
(obs.) A farmer.
(obs.) Some kind of bird, now uncertain.
(Xian.) A person dwelling upon the earth.
(now chiefly SF) A person from Earth.
(arch.) A materialistic person.
Earthy
(attest. a. 1400) Of or relating to earth, Earth, or (chem.) the earths, variously
Like earth; hence (minerol.) lusterless, friable, rough, &/or uneven or (elec.) similar in potential to the ground or (now chiefly pseudosci.) heavy, material; hence (esp. of humor) unrefined, crude; hence unpretentious, down-to-earth.
Made of earth; hence (now chiefly pseudosci.) restricted by its earthy composition, gross.
Covered with, full of, or containing earth; hence dirty.
On Earth; hence on or (esp.) in the ground.
(uncommon) Terrestrial.
Restricted to Earth; hence non-spiritual, material; hence worldly
Earthish
(attest. c. 1533; now chiefly SF) Earthly, of earth.
Earthite
(attest. 1814; now chiefly SF) A person from Earth.
Also earth-bred and -born; their latinate equivalents terrigenous and terrigenal; and, only dealing with things made of, like, or restricted to earth, earthen.