I think the best suggestion is to listen to speakers before making a purchase, since people's perception of the nature of sound is subjective, and speakers are the most variable component in the output side of recorded sound. Specifications are useful, but there's no way to discern from them how a particular speaker will sound to your ears.
With that said, quality powered studio monitors can be expected to be quite effective for use with PianoteQ, because generally speaking, monitors are designed to accurately represent sound over a wide dynamic range and frequency. That is important for reproducing the sound of a piano, and is where speakers like you've been using fall short.
The T20s for example don't even have technical specifications to give you any idea of their intended performance, but since they operate from an external PSU with a rating of around 22VA, they probably would claim about 8W RMS per channel. That amount of available power in a small speaker simply cannot provide undistorted sound loud enough to sound like a piano in full flight.
For your question about "music" not having the same distortion problem, I guess you're making a comparison with recorded music such as CD or other digital media files? That comes down to dynamic range, put simply the difference between the softest and loudest sounds being reproduced. Sound on, say, a CD is heavily compressed, so its highest level is not actually much higher than its average output, otherwise people wold be turning the volume of their radio/stereo/media player up and down all the time to hear the quiet bits then to not be deafened by the loud bits. Or a lot of modern music simply has very little dynamic range anyway, it's always just loud...
In comparison, a piano has a great dynamic range, between a single pianissimo note and the biggest double-handed fortissimo - when speakers are running at a level that makes average playing level useful, a fortissimo demands much more than the speakers can provide. Hence they are driven into distortion.
As a comparison, I use Event 20/20BAS V3 Monitors (with a Scarlett, although until it died I used my beloved eMU 0404USB), with 75W available continuous per speaker, and considerably more short term for transients. They are a good sounding speaker, yet not one I would necessarily suggest for PianoteQ, as they have a bit of a tubby upper bass region that works well with, say, electronica, and even orchestral like my favorite Saint-Saëns C Minor organ symphony, but needs compensation fiddling to sound completely even across the keyboard with solo piano.
Another thing to think about is whether you're wanting the speakers to give you the sound you will enjoy the most when playing live, or whether you're going to use PianoteQ to do recording, and are going to want to listen critically to those recordings and particularly mix tracks well. Quality speakers will let you do both of course, but for example there are monitors that are very revealing of detail and help in mixing precisely, but might not sound as enjoyable as a different monitor does when you're just having fun with PianoteQ!