GG5: an iron; 〔平ごて〕 a flatiron; 〔箱形の〕 a box iron; a goose.
WWW images
火熨斗	397

I think GG5's iron/flatiron is wrong.

@@ -16 +16,2 @@
-<gloss>a traditional ladle-shaped, charcoal-heated clothing iron</gloss>
+<gloss>traditional ladle-shaped, charcoal-heated clothing iron</gloss>
+<gloss>box iron</gloss>

https://ja.m.wikipedia.org/wiki/アイロン#炭火アイロン
「熨斗（『斗』はひしゃくでその形状をあらわす）」と言った。

日本大百科全書(ニッポニカ) confirms ladle-shape:
https://kotobank.jp/word/火熨斗-612051
小型の水柄杓(みずびしゃく)の形状をしていて

Previously-reported googits of 4million (still  1.4M today) may have been inflated (google returns 70 actual results for 火熨斗, 33 for 火のし):
火のし	682	63.2%
火熨斗	397	36.8%


Video history lesson on old irons(skipped to hinoshi section)...
https://youtu.be/ItmNablEsAg?t=84

https://kotobank.jp/word/火熨斗-612051

https://www.wafuworks.com.au/products/hinoshi-iron
This is a traditional kimono  hinoshi iron that was filled with hot charcoal. I've included an old drawing to show how they were used.  

https://chidorivintage.com/products/antique-c1900-japanese-charcoal-fire-iron-hinoshi-wood-handle-jk284
Japanese Charcoal Fire Iron Hinoshi Wood Handle

https://archaeology.jp/remains/kamiya/

Image search...  Images in kokugos (in kotobank). All links have pictures. 

2012 submission linked the "Iron" article from wikipedia and called it "an iron". It's the first iron listed in a chronologic history of all iron-like tools used in Japan.

Perhaps as a descriptive historic term that is not in practical use, 火熨斗 actually seems more prevalent in both yourei results and google results. 

斗 is the kanji for ladle.  the iron is roughly shaped like a ladle.  The kanji mean fire + unwrinkling + ladle.  We could make this entry slightly useful for kanji learners by making this association explicit in the gloss (as I have done).  Or something like a literal translation of daijs: "wood-handled, flat-bottomed metal bowl into which heated coals are placed, used for ironing clothes."

I was going to just say "traditional charcoal heated clothing iron", but even that is inadequate, as new-fangled devices used charcoal for a while longer. Such as this(unrelated, not a hinoshi):
https://chidorivintage.com/products/antique-c1900-japanese-sad-iron-charcoal-jk440

@@ -5 +5 @@
-<keb>火のし</keb>
+<keb>火熨斗</keb>
@@ -8 +8 @@
-<keb>火熨斗</keb>
+<keb>火のし</keb>
@@ -15 +15,2 @@
-<gloss>an iron</gloss>
+<misc>&hist;</misc>
+<gloss>a traditional ladle-shaped, charcoal-heated clothing iron</gloss>

http://ja.wikipedia.org/wiki/アイロン
4M hits for "火のし"

@@ -4,0 +4,3 @@
+<k_ele>
+<keb>火のし</keb>
+</k_ele>