JustSnilloc's answer is good and quantifiable. My answer is going to be more qualifiable and subjective.
Progress can be anything. If I weigh 350 lbs and I lose 100 lbs, that can be seen as progress. If I weigh 350 lbs and have 40% body fat but then lose 25 lbs and 10% body fat, that can also be seen as progress. Both are valid progressions and while one seems more drastic than the other (100 lbs is a bigger loss than 25 lbs) maybe reduced body fat is the goal vs reduce weight overall.
Common metrics can be aesthetic: weight lost, body fat percentage lost, or muscle growth; which one looks in the mirror, steps on the scale, uses calipers, etc and determines if they're going in the right direction. In your second example, the person gained weight (which could be good or bad because that weight could be muscle) but nearly doubled body fat percentage. Sub 15% is still good, but that person probably doesn't look shredded like they probably did two years prior.
Or they can be practical: Lifting more weight, more reps/sets. Typically the big goals I see online are to be able to bench 1.5x your weight, deadlift 2.5x your weight, and squat 2x your weight. If you are currently benching your bodyweight now, but several months ago you were benching only 75%, then there's progress (and you're now at about 3/4 of the way to meeting the big 1.5x). Your first example falls in line with this metric.
In summary, determine what you want to get out of the gym and find benchmarks to help you determine if you're going in the right direction. Do you want to get stronger? Figure out how much you can lift. After some months, if you're lifting heavier than before, you've made progress. Do you want to look better? Figure out your body fat percentage and figure out what the percentages are of the people you want to look like and shoot for that. After some months, if the percentage is going down, you're progressing.