Hi Jona - well, sorry you got off to an awkward start, but that was due to misleading information on your part. Your profile says Washington, many there grow their azalea bonsai outdoors all year round in the more coastal parts of Washington. So between the title and your first post, I certainly assumed the plant was outside. So I can see how the confusion began.
Many azaleas, especially Satsuki and other hybrids with the more sub-tropical species will set buds for the next years blooming, and are capable of blooming as if it were spring, with very few hours of chill. Satsuki azaleas bloom nicely even in Hawaii, where the 'winter chill' is only a 10 degree F drop for a few months from 70's & 80's everyday to 60's and 70's everyday. In the more temperate north, they go more completely dormant when outside for the winter. The parent plant that you took your cutting probably already formed its buds for next year, and bringing the cutting in the house signaled the cutting that it was spring. Not a problem, but you should keep it indoors for the winter. Next summer you can treat it more like an outdoor species, and leave it out all winter just as its 'parent' bush is outside all winter. If it is in a plastic pot, outside in winter is easy, several methods work, including just burying the plastic pot to the rim in a shady spot in the garden for the winter. If it is in a nice bonsai pot, winter it in a sheltered spot where temperatures stay between 32 F and 40 F (0 C to 4 C), so the pot won't be fractured by freeze thaw cycling, roots also need to be protected from freeze thaw cycling.
If it looks like the azalea cutting has a good set of roots, I would start leaving the terrarium open, to harden off the growth some. Step into it, initially just for a few hours, then withing a week or so you should be able to leave the top off completely, if the leaves will, return the cover to the terrarium. With the small soil volume of the pot, as it starts growing you will find out it can dry out quickly. Azaleas are a "I dry, I die" plant. You might want to repot to a larger pot soon. Being root bound is not a problem for azaleas, moving them to a larger pot is only necessary if you have trouble watering on the plant's schedule. Bigger pot, less frequent the need to water.
Give them as much light as possible indoors. They do not need as much as marijuana, but at least half what you would give reefer. Too dim, and growth will be elongated and weak.
Flowers are fun, and the whole reason to grow azalea. But they do put a high metabolic demand on young or weak plants. I would consider cutting the flower bud off to get the plant to shift to growing leaves and roots and more branches. But since it has already expended the energy getting the bud to its current stage, you could let it bloom, then after you enjoyed it a day or two, then cut it off. Don't let the bloom hang on for weeks. You want your cutting to work on growing.
About terminology - terrarium is good, I knew what you meant.
Cloning - for those in states where recreational pot is not legal, the word cloning is more generic than the word cutting. Cloning (propagation by creating clones) includes stem cuttings, leaf cuttings, rhizome division, grafting, meristematic tissue propagation, and other propagation techniques. Most of us when we hear the word cloning think of meristematic tissue culture propagation as this is the way the nursery trade, potted house plant trade, and orchid plant trade use the term. Cloning is so much shorter than spelling out "meristematic tissue culture", so the trades really use cloning to mean specifically that technique. Cutting is the term most commonly used for what you did with the azalea. Only in the marijuana trade is the word cloning used to mean "cuttings". Though most of us do eventually figure it out. Not a big deal, but since this forum includes people from all over the world, some are not native speakers of American English, assume most are not "hip" to the lingo, be tolerant if people don't understand your use of the terminology. And of course, bonsai as a hobby has its own "inside" lingo, that will take time to pick up. Words like nebari, shari, jin, uro, and whatever that damn word is for the first significant branch, all left me confused for a few years. And don't even get into the vocabulary for traditional bonsai displays. So we all try to be tolerant of lapses in terminology.
Welcome aboard, or as Source says, welcome to crazy. If you need more azalea tips, use the search function, we have a number of azalea freaks on this forum, you will find much good info the the Flowering Bonsai sub forum. Azaleas make great bonsai, and have a centuries old tradition all their own.