Weight/resistance training comes in many forms, as do strength and power sports. Some of those forms and the terms used to describe them, or describe particular training methods, have become a little confused and misleading, so this piece will hopefully untangle a few of them to help you make good training choices. There's no value judgement in this piece, I've done and enjoyed all these things and will happily use them all in the future, they're great sports. We're aiming for an understanding of them here, nothing more.

As a term, weight training can quite properly be used to refer to things like cross-training with weights, powerlifting, weightlifting, or bodybuilding. They could also be legitimately called resistance training and they often are, but "resistance training" encompasses any training where work is done against resistance and could include hiking with a loaded pack, swimming, aqua aerobics, spinning, rowing and many other things that aren't "weight training", so for the purposes of this piece, we're specifically discussing forms of weight training/sports, not general resistance training.

Bodybuilding is a sport in which the aim is to build as much muscle as possible while maintaining a balance of definition and symmetry. It's a sport that's often held up by the mainstream media or uninformed commentators as the immediate and inevitable outcome of weight training, a very unhelpful habit that hides the fact that bodybuilders apply a specific combination of personalised nutrition and consistent specialised training over years to achieve those results. For our purposes, don't worry about that, it wont happen to you unless you decide you want it to.
Despite not being our ultimate aim, bodybuilding is the source of quite a lot of very helpful science about building muscle and losing fat, something we can learn from. If you're an ageing alpinist, or one with a poor training history, or unhelpful genetics, you may need to build a little muscle at some point and it would be well worth learning about bodybuilding to inform that process. It's also quite common for female mountaineers to need to add some muscle at times do owing to different body composition and an unfortunately common lack of exposure to that kind of training in developmental years.
One element of bodybuilding that we may have little use for and should stay away from is doing lots of isolation exercises, doing a few different kinds of shoulder raises or barbell curls is fine if you're looking for a bodybuilding result, but for most of us they're a waste of time and energy if we're looking for added strength for mountaineering.

Weightlifting is a sport in which the aim is to lift the heaviest weight possible in two lifts, snatch, and clean & jerk. It's often called Olympic lifting and its through the Olympics that most people would be aware of it. For our purposes, this is one of the most influential sports as those two lifts use virtually every muscle we have and use them in such a coordinated way that doing them improves core strength, balance, body awareness, confidence, load carrying ability, metabolic fitness and speed, and makes a contribution to improving lactate threshold, VO2 max and injury resistance. Get some good coaching, perfect your technique (or tekkers, as you'll say in the gym) and these lifts will get you good results if you employ them in the right balance.
As with other methods, our aim is not to get to the point where we can lift as much as possible, unless you decide to take it up as a new sport, it's to lift enough to improve the markers I mentioned above and ultimately improve our performance on the hills.

Powerlifting is a sport comprised of three lifts - squat, bench press and deadlift. The aim is to lift the biggest combined total, though each lift has its own specialists and records. Powerlifting is another influential sport for us as those three lifts can target numerous muscles, the movements are functional, and as with weightlifting the coordinated involvement of joints pays off in a number of ways. Squats have an obvious benefit for leg and core strength, while deadlifts can offer huge improvements for hip and back strength in particular. Bench press uses every pushing muscle in your upper body and has an often underestimated level of core engagement. As above, if you're going to include these lifts in your training, get some good instruction and start light. They're great lifts and a very useful training method for us, but done badly they're potentially damaging.

Strongman/woman is a sport in which strength, power and skill are tested against a huge variety of weights. It's also known as Strength Athletics. Any given competition could consist of stone lifting, log press, deadlifting, farmer's walk, yoke carrying, flipping tyres or cars, truck pulling with a harness or rope, loading races and any number of similar events. Its relationship to alpinism is not immediately obvious, but some of those events are great ways to build strength and athleticism.
Farmers walk, in which you pick up two heavy objects and carry them over a course as quickly or far as possible, is great for leg and core strength, you can push any vehicle you have handy for a leg and glute workout and loading races, where you take a variety of heavy and/or awkward objects and move them over a course and onto a platform as quickly as possible, are as good a conditioning workout as you can find so long as you stick with an appropriate weight and safe objects.
These movements can be tough to learn as they can move through so many planes and angles under load, so start light, don't do them on bad surfaces and if you can find some coaching, take it. Personally I love these exercises, the variety is great fun and you can add them to strength workouts, interval sessions or circuits as you learn them.

CrossFit is a brand name that markets itself as a combination of training, lifestyle and community. As a system of training it is derived from the sports above, plus some additions from boxing, track and field and other sports, with some high intensity intervals thrown in, so there's really nothing new in it. CrossFit is often confused with cross-training, but the concept of cross-training existed for decades before CrossFit came along and just refers to training for your sport by doing something else, sprinting for speed skating, weightlifting for tennis, running for swimming etc. CrossFit codified that concept into a marketable, branded system and latterly evolved into a sport in its own right. It has come in for criticism owing to its purported injury rate and some of its coaching practices, but that's variable from one gym to another and avoiding injury is a function of understanding your own ability and movements.
Many fans of CrossFit will tell you it builds the sort of endurance we need for the hills on its own, but it doesn't and neither does the Crossfit Endurance offshoot, that's just a slightly longer version of the same thing. Both formats focus on "cardio" as though that was the whole story of aerobic ability. The improvements in cv fitness are an increase in vo2 max achieved through the HIIT elements, which is useful, but only one aspect of fitness. Focusing solely on those as either marker or training method would get you nowhere long-term and probably see you plateau and start to slide backwards in endurance terms fairly quickly.
As I've said before, there's nothing inherently wrong with Crossfit, so long as you go in with your eyes open and see it as part of your training, not the whole story. A session or two a week in the right phases of training along with longer aerobic training efforts like hiking, running etc could be useful, but you're unlikely to get benefit in the mountains from any more than that.

Besides the major sports and systems above, here's a few commonly used methods and moves it might be useful to understand;

Box steps are very applicable to mountaineering as they're an almost exact replica of the action of moving up a steep slope. They can be used as strength training by adding weight, in your hands, by wearing a weighted vest, or on a barbell across your shoulders. They're very versatile as they can also be used as local muscular endurance training by reducing the weight and stepping for time rather than sets of reps. You can reduce it further and step for to make it aerobic, but be warned, hours on a box step is mind-numbing.

Prowlers and similar sleds are a great way to work your glutes and your glutes should be worked hard, your arse is your engine!

Weight machines are often held to be an inferior method compared to free weights (barbells, dumbbells, kettlebells etc) and while that's true to some extent, there's really nothing wrong with machines if that's what you have available, or you're in the learning stages of training, or you want to train a movement without a training partner or spotter that might be safer on a machine.

Kettlebells have been around for a long time and have recently become popular again. They're a really versatile piece of kit and there are some great classes around. You can also find lots of ideas on how to use them on YouTube, search for KB workouts for strength, runners, MMA or, like this one, core. That video is from Redefining Strength and the channel is well worth following.

So, there's the major forms of weight training and sports and hopefully understanding the differences will help you understand how you might use them in the right phases of training, and how to avoid the pitfalls and myths that surround them.

Happy training,

Kevin.