What exactly is pickling?
Alright, I'm swallowing my embarrassment to finally ask the question I've google searched more times than I'd care to admit... I know what pickling is in terms of canning (soaking in brine/vinegar with spices for long-term storage), but I keep seeing chefs on Chopped say that they're going to pickle the beets, or pickle the onions, which to my definition, would take months. Is it just cooking in vinegar until soft?
Pickling is basically acidifying food. You can do this either by soaking the elements in vinegar (acetic acid) or in a salt brine which encourages anaerobic fermentation and the development of lactic acid.
What you're seeing on tv are "quick pickles." The porous vegetables are soaking up the vinegar brine and becoming sour within 30 minutes, so while they are not shelf-stable pickles, they are acidified.
I agree with the OP that pickling usually implies long-term preservation of something using vinegar and other ingredients. My process for onions takes a couple of days before they are ready for the Kilner jars and then it's about 3 months before they are ready to eat.
I think when TV chefs use the word these days, they basically mean they're going to marinate something in vinegar for a short time - usually a very short time to fit a TV programme schedule.
They are probably referring to a (relatively) short marinating in vinegar. I do a quick pickled onion this way - I slice the onions, sprinkle with salt and let them sit for an hour. Then I rinse off the extra salt, and soak them in vinegar, water, and pickling spices for a few days. The result tastes very similar to the long term version, requires less effort and storage space, and will keep for a few weeks with no problem. You can do the same with beets. My mom's on a low sodium diet for health reasons, and does onions and beets this way for salt free pickles.
I do a salad with cucumbers that's even faster - thinly sliced cucumbers and onions, salted and drained, and then dressed with vinegar or lemon juice and dill, and let sit for an hour or more. It comes in about half way between a salad and a pickle in taste and texture.
Pickling can be as simple as putting sliced onion in vinegar for 15-30 minutes before adding the onion to a salad or serving as a side to something. I've done that many times, and those pickled onions keep very well in the salted vinegar in the fridge for several weeks. Onions done that way tend to be quite sweet without that sharp edge that raw onions can have.
I do the same with beets, cucumbers, mustard leaves (the Chinese mustard leaves which are quite thick, although it could be done with thinner leaves, too.) For that matter, making sauerkraut or kimchi would qualify as pickling.
I make sauerkraut by slicing up cabbage and layering with salt and chilli powder (we like chilli powder in most things) and pounding a bit to soften the hard cabbage and encourage some of the moisture to leave the cabbage. The cabbage is then immersed below the liquid level with a touch of vinegar added as necessary. In a week on the counter, the sauerkraut is ready to eat and will store for months in the fridge.
I've also done pickles by canning, but honestly, I prefer the methods I described above. Canned pickles have no live bacteria, the good stuff. Using the methods above, the lactobacillus bacterias grow - that's healthy bacteria - and help to pickle the vegetables. Sauerkraut is supposed to be far healthier and easier to digest than cabbage prepared almost any other way. Other vegetables are supposed to similarly be healthier and easier to digest.
Ah, another form of pickling... Salted lemons, limes, or mangos. Take a lime and quarter. Coat in salt and stick it into a jar along with a bunch of other limes similarly treated. There you have pickled limes. Same can be done with lemons and mangos, and all are very good. Salted limes and mangos are used in Sri Lankan cooking, salted lemons in Moroccan, and probably a whole bunch of other cuisines, too.
Does this help?
I wouldn't put a whole lot of stock in what people on TV shows say about cooking. The vast majority of them don't know what many terms mean and know even less about food science and why food does what it does under certain circumstances.
In other words, just because someone on TV says they're doing "blank", doesn't mean they're actually doing it. You probably know more about food than most of those TV cooks.
Remember, they're there for for one reason: to attract viewers, not because of their skill or knowledge.