wow I sense many posts to come for this one
Yes, phrasing is definately crucial to playing musically. Just like in writing a story, when you're playing a piece, it gets split up into phrases (slurs, etc) paragraphs, main ideas (perhaps movements? ) themes, etc.
Playing musically is very hard to be defined, it's just like trying to explain to a colorblind person how red looks like. Telling them the frequency of the colour would do no good as it doesn't say anything at all about the colour, and using descriptive words won't do either.
I consider one's musicality as one's inner sixth-sense. Musicality is different in everyone, some might be more in-depth than others, some vary in style.
There are some definate required attributes to playing musically, that includes:
-Well control of dynamics
-Control of tonal quality, not too harsh, not too soft, just right
-Phrasing
-Showing variety and taste with the music, for example, in Alla Turca, if you are repeating the first theme twice, it is expected that you make either the first time softer or louder than the second time to show a little difference and variety (this is not a very good example but you get my idea)
-Adding your own taste to the music rather than copying all the musical styles of the great pianists on your recordings
-Emotion of the piece expressed, for example, you would try to bring out the melancholic attribute of some of Chopin's nocturne or the agitation of Beethoven's 3rd movement of his Moonlight Sonata or the brightness of Bach's 1st movement of his Italian Concerto
That's all I can rack up right now. There are many other things to musicality. One suggestion is to attend more recitals and concerts. One good way to sense musicality is to get a teacher and a student to play the same piece. If the teacher is good, you should be able to listen out some differences in their music making. Listen more to recordings. However, listening to recordings doesn't mean you copy every single thing from them. You can play it in the style of some other pianists as long as you don't play it note for note.
Btw, some drama might help too with your playing

like bouncing butts off the seat for fortissimo chords or swaying heads during legato sections
But don't overdo it.