Have you discussed BrE playing the piano/violin vs. AmE playing piano/violin?Not really, John, and it turns out that it's one of those things that's (all together now!) more complicated than you might think!
The Corpus of Contemporary American English (COCA) has 689 play* the piano to 309 play* piano. (The * there used as a wildcard in corpus searches; so play* gets us play, playing, played, etc.) That's more than two arthrous (fancy word for having a the) cases for every anarthrous (fancy word for not having a the) one--in American English.
British National Corpus, in comparison, has 14 arthrous cases and 1 anarthrous. (But keep in mind that the data from BNC is 20 years older than that in COCA.)
The moral of that part of the story: it would not be right to say that play piano is AmE for BrE play the piano. Instead, play piano is a lesser-used AmE variant of General English play the piano. The image here, from pianoplayingadvice.com, illustrates both variants living happily together.
Personally, I could say either, but prefer it with the the. A bit more rooting around in the Corpus of Historical American English shows a bit of anarthrous piano-playing throughout the 20th century, but it really gets going in the 1970s, when the proportions are like those in COCA.
But hold your horses. If we look at other instruments, it gets more complicated. (I'm rounding the numbers, unless they're <2 .="" comment-2--="">2>
- Violin: In COCA, the is favo(u)red 3:1. In BNC, 5:1.
- Harp: In COCA, the 4:1. BNC 8:0.
- Guitar: Ziggy played guitar. Maybe the Spiders from Mars made him do it without the the, but in 1990s UK, the British were following suit and, like 2010s Americans, using play guitar twice as much as play the guitar.
- Bass: Looks like a reversal! COCA 2:1. BNC: 1:5.
I tried discounting cases like playing (the) bass line/notes, but taking them out made no real difference.
- Trumpet: COCA1.4:1. BNC 5:2.
- Flute: COCA 4:1. BNC 8:1.
- Drums: Play drums outnumbers play the drums in both dialects. Is it because it's plural? But what about...
- Spoons: Tiny numbers, but more the in AmE and equal numbers of both in BrE.
Is it just with play, though? No. Going back to sticking with piano, COCA has half as many practic(e*) piano as practic(e*) the piano. BNC has four practis(e*) with the and one without.
On piano is also common in COCA (about 1/3 as many as on the piano). BNC has 20 on piano to 73 on the piano--very much the same. In this case, some of the on the pianos will have been about particular, physical pianos, as in I stubbed my toe on the piano. There's no possibility of I stubbed my toe on piano. But if a singer were giving credit to her band, she could say ...and Lynne Murphy on piano! or ...and Lynne Murphy on the piano! (Not me, of course, I only had a year of lessons.) I'm waiting for one of you to go out and listen to dozens of concerts with British and American singers to tell me if they all say on drums! on bass!
Finally, the why questions.
Why do we put a the before instruments? It's a funny thing. If I lie and say I play the piano, it's not a particular piano that I am playing. It's that I have the potential to play any piano. (Whereas if I say I've draped myself over the piano, it is a particular piano.) It's kind of like the bus in I ride the bus to work. In that case, it's not the particular physical bus we're talking about--that can vary. It's the whole package that goes with bus-riding. I ride a bus that travels along the route between my street/road and my workplace. There's a package that goes along with pianos too. I'm not just playing the instrument, I'm playing music on the instrument. The music that I know how to play on any "the piano" is kind of like the routes that I travel on any "the bus".
In spite of all that, there's no pressing semantic reason for the the. We don't play the cards or play the dominoes even though similarly, if I say I know how to play dominoes, I'm saying that I know the rules for playing on any instrument of that type (any set of dominoes). [Yes, dominoes are the instrument, not the game--though people who only know one domino game tend to call it 'dominoes'. I am particularly fond of Mexican Train.] So why do we usually have a the with musical instruments, but not with game equipment? (The answer: because that's what we learned to do.)
The arthrous version is unhelpfully ambiguous, so maybe that is a contributor to the rise of the anarthrous alternative. If I say I play the piano I could be trying to point out that I know how to play a piano (so invite me to play at your wedding), or it could be saying that I play a particular piano habitually (so don't get rid of it). I play piano doesn't seem to have that ambiguity, so could be seen as more communicatively efficient. The play + bare-noun construction is familiar, since we say things like I play tennis, I play jazz, I play goalie.
If you want to carry the conversation toward(s) other cases of (an)arthrous variation in AmE and BrE, have a look at the past posts with the 'determiners' label. I've written about some of the famous ones already, and your comments on them would be most welcome at those old posts (which are still regularly read). And you're most welcome to carry on the conversation about musical instruments (and games) on this post, of course!