Yes, of course there is a market convention.
We can try to imagine how this worked in the 19th Century. The bond belongs to Mr. S, a wealthy capitalist. On 10/03 Mr. S is legally entitled to receive a coupon payment. So the first thing he does (of course) is he goes down to the US Treasury office on Wall Street, and shows the bond. The clerk "clips" (cuts out) the coupon from the bond, returns the bond to the owner (after stamping "coupon paid 10/03/1819" on the back) and gives the coupon payment in cash to Mr. S.
Having enforced his right Mr. S now has to live up to his obligation: he sold the bond to J.P. Morgan recently and is now (10/03) obligated to settle the sale. So he takes the bond (what's left of it, i.e. without the coupon) over to the offices of J.P. Morgan across the street. There he gives the bond and receives the price that was agreed. (Mr. Morgan, the buyer, is not surprised that the coupon is missing, he would have done exactly the same thing)
The modern convention is similar: when the bond settles on the coupon date, the coupon is paid to the seller, and the invoice price is equal to the quoted price (i.e. there is no accrued interest). The buyer is entitle to the next coupon (all of it or a part, if he sells it early) but not today's coupon.
(This is for US Treasury Bonds, in the UK it is more complicated).