The processes used in terraforming lifeless planets were pioneered not by the Bright Way, but by the Lifebringers, a sect of Neoshamanists. Whereas the Bright Way sought sapient life dwelling among the stars, the Lifebringers attempted to recreate sapient life anew through artificial selection. They never achieved their goal, but did contribute tremendously to the fields of biology, ecology, medicine, and planetology along the way.
Once the other rocky planets of Focus were found to be devoid of life, the Bright Way lost interest in them, leaving the Lifebringers to go to work.
The foundation of the terraforming process is a complex microbiome selectively bred from existing species of extremophiles and chemotrophs on Yih. These come packaged together in something known as "rock eater".
Different strains of microbe have different metabolic processes that contribute ultimately to oxygen formation and (perhaps not yet cannon) soil and water formation. In the early stages, a positive feedback loop is initiated whereby the more oxygen is produced the more certain species within the microbiome flourish, creating even more oxygen, however, other species that the oxygen producers depend on are anaerobic, and die after the partial pressure of O2 reaches breathable levels. This causes the oxygen producers to die in turn, and most of the rest of the microbiome collapses, leaving only the foundation on which natural life can subsist, completing the terraforming process.
How livable conditions are maintained afterward depends on the size of the body and its distance from Focus. Larger bodies can maintain the atmosphere through their own gravity assuming a self-generated magnetosphere is either already present or able to be kickstarted by Science™️ (see the movie The Core for what I'm thinking here).
Smaller bodies require active intervention to maintain these conditions. The Science™️ in this case is generated by orbital infrastructure which requires upkeep. The upside to this is that planet-wide climate control is possible, and the local weather service doesn't predict the weather, they cause it. Moons and dwarf planets in the Outer Belt are nominally able to maintain an ambient temperature that humans would probably find "brisk" or "nippy". (Also tentatively cannon, the populations of these planets have developed thicker fur to compensate, creating a distinct floofy Outlander phenotype.) Penny-pinching politicians sometimes decree that the global thermostat be turned down to "bone-chilling" to save money.
Bodies outside the orbit of Moonlitter are too far away for Focus to provide meaningful illumination, so more orbital infrastructure provides this light along with surface lighting.
Larger bodies can't be actively climate controlled, and have their own more or less naturally evolving atmospheric processes as seen on Earth and Yih. Hearthside has artificial aerosols in its atmosphere that reduce surface insolation to livable levels. (Definitely already cannon is that Hearthsiders have evolved larger ears to dissipate heat.)
Yinrih don't discover FTL in the form of the mass router until after First Contact with humanity, dozens of millennia after every suitable body at Focus has already been terraformed. Prior to the mass router interstellar travel is limited to missionaries of the Bright Way searching for worlds that already have sophonts, a very risky endeavor that they only undertake out of zeal.
Post FTL, ethical questions come to the fore, and the Bright Way takes a dim view of being grabby. There's internal disagreement on which planets qualify for this designation. Some say all exoplanets are off limits, others say only those with promising prebiotic environments should be left alone. The doctrine as written seems to preclude only planets in the habitable zone of their star. As far as secular law is concerned, if you maintain a presence in a planet's gravity well it's yours, and the Bright Way's missionary efforts partially pivot to squatting on such promising planets to prevent others from bulldozing the environment.
At this point the Neoshamanists are much more homogenous, with both the Lifebringers and the Mindseekers (who sought to create strong AI and likewise failed) having long faded into history. They may or may not care one way or the other at this point. The setting doesn't really explore far beyond present day.