Yes, that's true. You need to be comfortable with your skating and focus on your skating technique so you don't fall.
Dancing on the floor can be very improvisational, but dancing on ice needs to be planned carefully. ESPECIALLY if you're skating with a partner, you need to know exactly what you're going to do next and exactly what your partner is going to do next.
That's why ice dancing starts out with set pattern dances. Everyone does exactly the same steps. (In many of them, including all the beginning dances, both partners do the same steps side by side in Kilian position. In a lot of the more advanced dances, each partner is doing something different, but everybody who skates the "lead" steps/men's steps will skate exactly the same steps, and same for everyone who skates the "follow" or women's steps.)
You have to learn the correct steps with the correct technique and correct timing. You have to have good control of the actual skating before you can start to really dance to the music. Doing it correctly is very important for your safety, and for the safety of the other skaters on the ice.
If you like to move to music and want to move to music on the ice, stick to gliding on two feet and just move your arms or maybe your shoulders, etc., in a controlled way, only as much as you're comfortable with to maintain your balance on two feet.