I was, still am, a tense student. Let me tell you, while nerves for the lesson may be a problem for some, at the root of it talking softly and being polite is not going to solve the problem.
First, let me tell you a little bit of my background. I was my class's salutatorian. This means I strove to do my best in every class, took all of the advanced classes, and, though I didn't show it to people, poor grades always left me depressed and scared about my average. All through middle school and high school I did Select Chorus and Jazz Ensemble, and I also tried tennis, track, and cross-country. There were days when I went to school at 7:35 am and left at 7:30 pm. On Saturdays I was in a bowling league and, being a competitive guy, I liked to win and strove to bring my average up to a 180 and manage to roll a 600 series. Since I was eight I played piano. My first teacher for five years held recitals, and people always thought I was great (no prodigy, though). Then, after a stupid fight, I switched teachers and there were another three before I went to my teacher for high school. The bad habits and tension were already ingrained before that because the teacher before her loved to put difficult pieces on me, and he taught musicality rather than motions, not that he taught much at all. I was also in my church choir. So, all throughout high school I had Friday nights off, and the weekend. Mon - Thurs there was something going on until at least 7:30 every night. I tell you this not to impress you, but to impress upon you that I was an overachiever. I still am, but I'm trying to loosen up.
For me, tension is a deep-seated problem. My three big areas that my teacher noticed are my shoulders (always up), chest (caved in and tight), and my jaw (locked, doing weird things while I play). Partly because of my tension, and partly because of my lack of solid instruction, I also had great technical deficiencies, which furthered my tension problem. Now, I closely monitor my breathing and blinking during practice, I look for tension in the hand, forearm, upper arm, even my legs tend to cramp up when I play. I also try and do yoga daily. I've made progress, and I'm confident my teacher can help me clear away the problems, but I work at relaxation every time I sit at the piano. It's gotten to the point where I don't even realize a muscle is tense until it starts to hurt. I have horrible posture, but when I roll my shoulders back, I find that correct posture is very uncomfortable and hard to maintain, because my chest is so used to being closed and tight. It actually takes effort to relax.
So, go, I ask you to please, please keep investigating this. What you say worries me, because you are a natural, and I'm glad for you, but that also means that you might not understand the problem. I'm not saying that tension is forbidden, but I do think that as a teacher you must watch for it like a hawk (pardon the cliche), because I don't want your students to end up like me. You might look into Feldenkrais instruction for musicians. I can't take it yet at my school, but I've worked with a certified instructor (other than my piano teacher). She was the one who spotted immediately that I hardly breathe, blink, or swallow when I play. There's also the Alexander technique, which is different but has the same aims. I'd say you, as the teacher, are the most important influence in helping your students relax. I've read a lot of piano material, but none of it truly makes sense until my teacher explains and demonstrates the material in her way. Then I can connect it to my reading, but instruction comes first.
I don't know how common tension is as an issue, but it is definitely a serious one.