Is strength training only lifting weights?

Yes

I would say that you can count bodyweight stuff as strength training as long as you up the intensity/ add weights in time. Most people start with bodyweight training, then progress to weights.

There are many ways to keep it challenging (try pistol squats, for example).

I feel silly having to ask this, but what exactly counts as strength training?

I've been doing a couple pilates and les mills bodyflow classes, where you're using your own bodyweight as resistance and whatnot, so can that count as strength training?

bodyweight can be used for strength training. but not in the examples you listed..

strength training is maximal load compound movements where you hit total muscle failure in ~5 reps or less.

strength training training achieved by working dynamically against high loads or statically against fixed resistances. In both cases the forces involved must be such that relatively few repetitions are possible without a substantial rest period. A sustained strength-training programme will progressively increase the loads and number of lifts over a period of months, the exercises being performed in several sets, each embodying a specified number of repetitions. In the first 2-3 months the main improvement in strength is attributable to increased recruitment of motor units within the pre-existing muscle mass (the 'neural phase' of strength training); thereafter, increase of muscle fibre diameters is the major factor ('hypertrophic phase'). aka resistance training. See also weight training.
Dictionary of Sport and Exercise Science and Medicine by Churchill Livingstone © 2008 Elsevier Limited. All rights reserved.

Body weight training can be strength training. But there has to be a progressive format to it. Those classes don't have that.

definitely this

To expand on what progressive means, basically this in a nutshell:

As you get stronger, you need to do something to continually increase the difficulty of the exercise so that you keep challenging your strength. And no, simply adding more reps doesn't cut it. You need to make the exercise itself more challenging while keeping to roughly the same rep range. As long as you are able to continually do this, you are able to continue to grow stronger, and thus, it is strength training. Bodyweight based strength training can, depending on your perspective, be either the most efficient or the least efficient. The reason is because, with bodyweight work, progressing doesn't just mean increasing the load. It means continually moving on to variations of the exercise that are more challenging for not only your strength, but also your balance, flexibility, coordination, and control. Some see this as a good thing, because it means that you need to focus specifically on your weaknesses before you can progress, leading to a more well rounded base of fitness. Some see it as a bad thing, because it means progression itself ends up coming at a much, much slower pace than simply being able to pile more weight on that barbell.