What you described is meditation in its purest sense, as it can take many forms. The true definition of meditation is mindfulness. If you do anything with mindfulness you are meditating.
For me I practice sitting meditation, walking meditation, and yoga. Creating art, painting and photography, are also both highly meditative experiences that I enjoy immensely.
I've made great effort to meditate while coding, but I have found it to be nearly impossibly as my mind needs to change its pace in order to solve problems. Keeping with mindful breathing while coding is possible and very beneficial.
One of the greatest benefits of meditation, as you mentioned, is for problem solving. Solutions come when you 'allow' them to. Walking meditation is very effective for this. I keep a timer and take a 5 minute walk for every 25 minutes of work.
Creating art, painting and photography, are also both highly meditative experiences
They're certainly mindful, but I like to distinguish the mindfulness of meditation from the mindfulness of concentration on something specific.
When you're concentrating on something like coding or creating art, the thoughts themselves provide the framework of your mindfulness and drive your mind's focus.
When I'm meditating, I try to create that intensity of mindfulness without the framework of something specific to concentrate on. By learning to separate the focus from the things that I focus on, I feel I'm doing something more fundamental with my brain.
I guess the people who've asked me must not know much about meditating. I admit I only have a superficial knowledge of it, as well. The people I've talked to see meditating more of a 'relaxing' experience, I guess.
This is very true, as it can be hard to sit or focus, and it tends to bring up many uneasy thoughts and emotions. But it is very helpful to find a place of relaxation, since meditation itself is simply a vehicle for self discovery. Meditation is the journey, and the journey is the destination.
The ultimate "goal" is transcendence, which untethers your consciousness from your physical body and is quite a liberating experience.
The catch to it all, is that you can't have a goal while meditating, other than to be present in the moment, which is an admirable goal in itself.