Have you ever seen a bad actor perform? Someone who has memorized all of their lines and goes through the appropriate motions on stage, but you just don't feel anything from their performance?
Imagine taking a script from a play showing a highly emotional scene -- say, an argument, or a romantic scene. Now imagine reading all of the lines in a monotone -- you observe all the commas and periods and pronounce every word clearly with good diction, but otherwise you put no rises or falls in your voice, no subtle changes in speed or inflection. You read just like a robot. But, you follow the script meticulously. When you see a stage direction -- "(turn to her)" or "(puts arm around him)" or "(points at her)" -- you perform the action requested, but do so deliberately and mechanically, as a robot might.
Would this be a good performance of that emotional scene? Probably not. A script for a play simply can't contain all the details necessary to bring the words to life. There are the words, and punctuation, and suggestions like stage directions, but the details need to be filled in by an actor to transform the blueprint on the page into a living, breathing human performance.
That's exactly like a musical score. The notes on a page are not "music," even though we often call them that. They are a blueprint, just like a script to a play. Yes, you need to play the right notes. Yes, you need to hold them the correct durations. Yes, when the score says to play louder or softer, you should do so. But doing all that correctly can still be like the robotic monotone reading of a script. Maybe you hold some notes a couple milliseconds longer and others a couple milliseconds shorter. (This is called microtiming, and recent studies show skilled performers make a lot of such miniscule adjustments, even as they are "playing what's on the page.") Maybe when it says to get louder, you create a phrase by gradually emphasizing certain notes. Maybe, depending on your instrument, you make subtle changes in timbre or tone or articulation at moments that enhance the performance.
Suddenly a string of 4 bars of individual notes becomes a phrase. 16 bars of individual notes are shaped into a theme. It's no longer a robotic monotone execution of the script, but a living romantic scene, or an intense argument, or whatever.
You seem to think that you need to gesticulate wildly or move your body around to show "feeling." But you said, "think of a concert pianist," and the first person I always think of is Vladimir Horowitz. There is someone who played with intense "feeling," but his head and shoulders and torso would be very quiet and at ease, while his hands were flying around the keyboard doing the most amazing things. He wasn't a showy performer like some pianists, but his interpretations brought the music to life. For music, it is about the sound and the technique -- you don't need to flap your arms around or rock about like you're in some sort of religious ecstasy. But producing a good sound and good technique in a musical performance is more than simply playing "what's on the page."
Take note of that last sentence. What your judges in your exams were really telling you is that you weren't playing musically. Music is about creating a flowing experience in time, about connecting notes and phrases and themes, about sensing a coherent interpretation that transforms the "script" into the "romantic scene." But saying you weren't playing "musically" is vague and perhaps insulting. So people talk about playing "with feeling," because the process of interpretation for music is much like telling the actor who is reading the script during the love scene that it needs to be performed with more sense of emotion/feeling. It needs to be acted, to be interpreted, to have a sense of more than just a collection of words -- to feel like a scene.
A piece of music is the same. Some performers literally do experience intense emotions and feelings as they perform, which help them to interpret music. Others recognize the subtle interpretive decisions necessary to bring the music "to life" even without invoking emotions in themselves. Just like different kinds of actors -- some who really want to feel and inhabit the character they are portraying, while others are bit more "distanced" about their roles -- there are different kinds of performers. If flapping your arms around helps you to "feel" the piece start to come together, maybe you do that. Some performers find that helpful; others don't.
But the point is transforming the musical "script" on the page into the nuanced, musical experience of sound moving through time. The notation is only a rough blueprint, an approximation, a starting point. Like an actor, you need to figure out how to "bring it to life."