Timekeeping Systems on a Habitable Moon

Question

Ok, this is take two. Hopefully this is specific enough. I have a link to the original question here.
And for those of you eager to see my sources for this, I'll link two videos here from the YouTube Channel Artifexian.
Worldbuilding: Terresstrial Moons
Worldbuilding: Gas Giant Moon Systems and Habitable Moons

I have a 6-planet solar system (only three planets have been fleshed out so far), with a purple gas giant in the Habitable zone. It has 3 Earth-ish sized moons: one small ice-covered ocean moon (like Europa), one swampy rainforest moon (like Endor/Dagobah), and one Desert moon with 60% of its surface covered in dry land. The desert moon's 'year' is 147 Earth days long, but due to the moon's rotation, it's actually more like 117 days long.
(I'll put stats and numbers in comments, if any of you wanna recreate it in Universe Sandbox or stuff then let me know.)

An intelligent species will inhabit the desert moon, the furthest one out. My big question is what are some possible ways they can create a calendar?
I've considered making the year based around the moon itself, but is it possible that they could see that their planet (the big, bright purple thing in the sky) orbits the sun (the big REALLY bright thing in the sky)? Or maybe they can tie time periods to the other two big moons?

(P.S. The moon orbits at a sufficient angle and distance that regular eclipses aren't a thing, so if anything, solar and lunar eclipses would be about as regular as Earth's eclipses.)

.....

Edit: here's the planet and moons' mechanics, per request. Units are given in terms of Earth. (Except for the planet, whose mass and radius is given based on Jupiter.)

Planet 2: Purple Habitable Gas Giant
Dist: 1.2 AU
M: 2.15 Mj
R: 2.89 Rj
g: ??? m/s^2
d: 0.88 g/cm^3
Orbital characteristics:
a: 1.2 AU
b:
e: 0.0038
i: 0.00028°
Ω: 83.2°
ω: 335°
Year: 1.3 Earth-Years
Moons: 7 (3 Earth-Sized)

Moon 1: Ice World
M: 0.74 Me
R: 1.01 Re
g: 6.93 m/s^2
d: 3.85 g/cm^3
Orbital characteristics:
a: 2,000,000 km (0.01336 AU)
b: ??? km (0.0 AU)
e: 0.027
i: 1.72°
Ω: 152°
ω: 20°
Year: 0.57 days (12.5 Earth Days)
It's icy because it formed with the gas giant out in the distance. It lacks any form of greenhouse gasses, and the surface ice reflects sunlight away, keeping it from melting fast enough. Tidal forces from the planet, though, keep it warm internally, enough to support a subterranean ocean.

Moon 2: Endagobah (Swamp World)
M: 0.76 Me
R: 0.92 Re
g: 8.85 m/s^2
d: 5.40 g/cm^3
Orbital characteristics:
a: 5,161,126 km (0.0345 AU)
b: 5,410,000 km (0.03 AU)
e: 0.0037
i: 2.45°
Ω: 98.5°
ω: 264°
Year: 42 days (55.4 Earth Days)
This place has bugs. Nobody'll go there anytime soon.

Moon 3: Desert World
M: 1.36 Me
R: 1.103 Re
g: 10.99 m/s^2
d: 5.59 g/cm^3
Orbital characteristics:
a: 10,500,000 km (0.0702 AU)
b: 10,499,112 km (0.0718 AU)
e: 0.013
Periapsis: 10,363,500 km (0.069 AU)
Apoapsis: 10,636,500 km (0.0711 AU)
i: 3.67°
Ω: 160°
ω: 98.0°
Year: 117.6 Days (147 Earth Days)

Answer 1

Just like there are multiple ways we could define days (solar vs. sidereal), there are many options available to the inhabitants.

Solution 1: Bright=Day, Dark=Night

Considering we base our days around the "bright/dark" cycles caused by the Sun, it is likely the moon inhabitants would as well. Assuming the host planet does not obscure the star for long portions of time, e.g. while the planet is between the moon and the star, the inhabitants will likely define a day similar how we have, i.e. start timing when the star is the highest in the sky and stop when it has returned to its highest point. This has many advantages. It ties their days to the brightest object in the sky (and most obvious). It likely ties the days to their circadian rhythm. It avoids unnecessary corrections to the days through leap hours or such. This seems to be the most intuitive, but there are other options.

Solution 2: Relative to the Planet

Now they could do the same thing, but using the planet, however this would cause the brightest light source (the star) to change it's height in the sky until its pitch-black at "high noon", when it was originally brightest. Or they may be tidally locked in which case the day would be infinite, which might be a problem for a work day. I would likely rule this out do to practical purposes.

Other Solutions

If they could observe the rotation of the host planet, they could define a day based on the host planet's rotation (or day) relative the star, but this is very difficult, since they would have to know the relative position of points on the planet to the star, independent of their own location in the lunar orbit. Precisely mapping the surface of the gas giant, or even the fact that it is rotating would be very difficult. I would rule this out due to the sheer difficulty compared to other options.

Determining the Heliocentric Model

It likely wouldn't be too difficult to determine that they are orbiting a planet which is itself orbiting the star, barring religious backing of certain moon-centric viewpoints. Since two gravitationally bound objects orbit their center of mass, there can be symmetrical views of which object is orbiting the other, and the case reduces to that of our solar system. Some intrepid astronomer(s) will eventually map out the relative positions of background stars, the host star, the host planet and the other moons over long periods of time and, given a rudimentary knowledge of orbital mechanics, would recover the correct answer.

Months and Years

Once you have solar days (using the dark and bright cycles caused by the star), it is easy to define a month from the rotation of the moon around the planet (or planet around the moon the inhabitants might believe) by observing changes in the planet's position in the sky relative to the star. Years would be more difficult, but they could be determined by recording the motion of the background stars in relation to the host star. Mark the constellations visible at midnight on a certain day, and their positions. When they return to those same positions, you know that the planet has completed an orbit around the star.