I think that's a good idea. I've drafted an edit for the タンヤオ entry.

I think we should. In my experience it's definitely the most commonly used name among English-speaking players.

If we're going to call it a "tanyao" then that gloss should be in the actual tanyao entry.

https://riichi.wiki/Kuitan
  'Kuitan 「喰い断」 is a rule allowing open tanyao. If kuitan is disallowed, then it would be specified as "kuitan nashi", and tanyao would be restricted as a closed only yaku.'

Daina Chiba, Riichi Book 1 (page 11; https://dainachiba.github.io/RiichiBooks/)
  'Games in the first row (喰断ナシ kuitan nashi) are unusual games where open tanyao (All Simples) is not allowed; you have to have a concealed hand to claim tanyao.'

jitsuyou:
  麻雀で、チーやポン、カンなどをしてタンヤオの上がりに向かうこと。鳴きタンともいう。

wikipedia (https://ja.wikipedia.org/wiki/断么九#クイタンのありなし)
  副露したタンヤオはクイタン（喰いタン、など）と呼ばれるが、クイタンを認めるかどうかは地域やルールによって異なっている。 

〈喰（い）／食（い）／くい／クイ〉〈断／たん／タン〉

Google N-gram Corpus Counts
╭─ーーーー─┬─────────┬───────╮
│ 喰いタン │ 110,809 │ 84.2% │
│ 食いタン │   9,978 │  7.6% │
│ 喰いたん │   1,887 │  1.4% │
│ 食いたん │     911 │  0.7% │
│ 食い断　 │     627 │  0.5% │
│ 喰い断　 │     388 │  0.3% │
│ 食たん　 │     343 │  0.3% │
│ 喰タン　 │     207 │  0.2% │
│ 食タン　 │     111 │  0.1% │
│ 喰断　　 │      80 │  0.1% │
│ 食断　　 │      39 │  0.0% │
│ 喰たん　 │      25 │  0.0% │
│ くい断　 │      20 │  0.0% │
│ クイ断　 │       0 │  0.0% │
│ クイタン │   3,018 │  2.3% │
│ くいたん │   2,648 │  2.0% │
│ くいタン │     370 │  0.3% │
│ クイたん │      72 │  0.1% │
╰─ーーーー─┴─────────┴───────╯

No need to explain what tanyao is in this entry.

Despite what the first reference says, the Japanese sources say that "Kuitan" doesn't necessarily refer to the rule itself. The rules are either 喰断アリ or 喰断ナシ, and the former is sometimes abbreviated as 喰断.

@@ -4,0 +5,18 @@
+<keb>喰いタン</keb>
+</k_ele>
+<k_ele>
+<keb>食いタン</keb>
+</k_ele>
+<k_ele>
+<keb>喰いたん</keb>
+<ke_inf>&sK;</ke_inf>
+</k_ele>
+<k_ele>
+<keb>食いたん</keb>
+<ke_inf>&sK;</ke_inf>
+</k_ele>
+<k_ele>
+<keb>食い断</keb>
+<ke_inf>&sK;</ke_inf>
+</k_ele>
+<k_ele>
@@ -5,0 +24,5 @@
+<ke_inf>&sK;</ke_inf>
+</k_ele>
+<k_ele>
+<keb>喰断</keb>
+<ke_inf>&sK;</ke_inf>
@@ -12 +35 @@
-<re_nokanji/>
+<re_inf>&sk;</re_inf>
@@ -16 +39 @@
-<xref type="see" seq="2749690">断ヤオ九</xref>
+<xref type="see" seq="2752740">タンヤオ</xref>
@@ -18,2 +41,2 @@
-<gloss>having an open hand and no end or honor tiles (meld; not allowed in some rules)</gloss>
-<gloss>having only suited tiles between 2 and 8 inclusive and having called tiles</gloss>
+<gloss>tanyao with open melds (not allowed in some rules)</gloss>
+<gloss>open tanyao</gloss>

@@ -18 +18 @@
-<gloss>having an open hand and no end or honor tiles (yaku, not allowed in some rules)</gloss>
+<gloss>having an open hand and no end or honor tiles (meld; not allowed in some rules)</gloss>