My family uses that rule; in fact I think it's in our rule book and applies to some other contested discards.

@@ -20 +20 @@
-<gloss g_type="expl">house rule specifying that in the event of two or more players declaring a win from a discarded tile, the only valid win belongs to the player closest (in turn order) to the player who discarded the tile</gloss>
+<gloss g_type="expl">house rule specifying that in the event of two or more players declaring a win from a discarded tile, the winner is the player closest (in turn order) to the player who discarded the tile</gloss>

I chose "optional rule" instead of "house rule" because I thought maybe the latter might be too jargony, but maybe I only made it more confusing. The option doesn't belong to the players at the time of payment; whether or not atamahane is in effect is decided before the game begins. So it wouldn't be correct to say that the rule allows the payer to do anything. The rule specifies to whom the payer must pay.

There's only one payer (the player who discarded the tile).

Maybe this is better? I think only one comma is needed here, but I'm doubting myself a little bit.

@@ -20 +20 @@
-<gloss g_type="expl">optional rule used when two or more players declare a win from a discarded tile, allowing the payer to only pay the player closest (in turn order) to them</gloss>
+<gloss g_type="expl">house rule specifying that in the event of two or more players declaring a win from a discarded tile, the only valid win belongs to the player closest (in turn order) to the player who discarded the tile</gloss>

the previous gloss was an explanation, but not an explanatory gloss as we currently define them - if the word is an noun, the expl gloss should also be an noun. 

should it be "allowing payers" rather than "allowing the payer"? Can there be more than one payer?

@@ -19,2 +19,2 @@
-<gloss>atamahane (optional rule)</gloss>
-<gloss g_type="expl">in the event of two or more players declaring a win from a discarded tile, payment is made only to the player closest (in turn order) to the payer</gloss>
+<gloss>atamahane</gloss>
+<gloss g_type="expl">optional rule used when two or more players declare a win from a discarded tile, allowing the payer to only pay the player closest (in turn order) to them</gloss>

jitsuyou
https://kinmaweb.jp/mahjong-rule/atamahane
https://riichi.wiki/Atamahane


Google N-gram Corpus Counts
╭─ーーーーー─┬───────┬───────╮
│ 頭ハネ　　 │ 2,324 │ 44.0% │
│ 頭はね　　 │ 1,907 │ 36.1% │
│ 頭跳ね　　 │   886 │ 16.8% │
│ アタマハネ │    98 │  1.9% │
│ あたまはね │    70 │  1.3% │
├─ーーーーー─┼───────┼───────┤
│ 上家取り　 │   348 │  N/A  │
╰─ーーーーー─┴───────┴───────╯