OK

I think this is sufficient.

@@ -12,2 +11,0 @@
-<xref type="see" seq="1056690">サタン</xref>
-<xref type="see" seq="2856037">ルシファー・1</xref>
@@ -15,2 +13 @@
-<gloss>fallen angel (oft. Satan, Lucifer)</gloss>
-<gloss>demon</gloss>
+<gloss>fallen angel (esp. Lucifer)</gloss>

Note that the field tag applies to 堕天使; not the gloss.  I think the "demon" was OK.

@@ -15,0 +16 @@
+<gloss>demon</gloss>

daijs.
キリスト教で、悪魔のこと。神の試練に堪えきれず下界に堕おとされた天使、特に、その首領ルシフェル。

nikk says:
キリスト教で、悪魔をいう語。神の怒りを買い、天上界から下界に堕落させられた天使。

To interpret nikk, Wikipedia says of 悪魔:
https://ja.wikipedia.org/wiki/悪魔
キリスト教神学では、神に対して謀反を起こした堕天使（サタン）の手先である。
In Christianity, 悪魔 are Satan's henchmen(one of our many glosses - demons).  So nikk isn't singling out Satan at all (as far as I can see).
And Satan is, of course, a *specific* 堕天使 (神に対して謀反を起こした堕天使). 


堕天使	194658	41.8%
ルシファー	68682	14.7%
サタン	202355	43.5%

Apart from the main comments below, if you do an image search for 堕天使, you'll quickly see that this has a "romantic" use in Japanese imagery that seems somewhat disjoint from actual Christianity.  堕天使, like 天使, seems like a fun concept to play with (like medieval knights) without being overly invested in any particular religious principles. In these cases, I think "fallen angel" alone still remains the best gloss. 

===
I got this term from jisho some years ago, and the gloss collection has always seemed off to me. I'm suggesting what I think is a "better" way to do this. 

My core motivation is that "fallen angel" is essentially a perfect 1-to-1 gloss.  If you want to translate "fallen angel" into Japanese, this term is perfect.  If you want to translate 堕天使 to English, "fallen angel" is perfect.  Everything other gloss that was here just muddies things up. (Lucifer <-> ルシファー, Satan <-> サタン, demon <-> 悪魔)

Almost every reverso gloss is "fallen angel". Why wouldn't it be? And where it isn't, it is never glossed as "Satan" or "Lucifer". 
https://context.reverso.net/translation/japanese-english/堕天使

daijs just says: "特に、その首領ルシフェル".  "Often refers to the head of the fallen angels, Satan" could be said just as easily of the English term "fallen Angel".  But context is always required.  You can never blindly change 堕天使 to "Lucifer" unless you understand the text.  If you understand the text, that's your prerogative.  If you don't understand the text, "fallen angel" is always safe. 

daijs' mention 特にルシファー is part of the *definition*, to provide information to a Japanese reader about the meaning of the term.  But that doesn't make ルシファー an appropriate gloss.

The number one pinned gloss for 堕天使 on weblio is "seducer".  Another on the page is "tempter".  I wouldn't suggest adding those either. 
https://ejje.weblio.jp/content/堕天使

For comparison, eow.alc.co.jp (usually pretty good) has exactly one gloss:
https://eow.alc.co.jp/search?q=堕天使
fallen angel

oft. and not esp. because:
1. I'm suspicious about "fallen angel" usually being a reference to Satan in the first place.  Growing up Catholic myself, "fallen angels" was usually plural. Satan has perfectly serviceable names of his own.
2. Based on google search, I think 堕天使 is extremely common in pop-culture to refer to something that is maybe satan-like, but not really part of Christianity.

[fld=Christn] is a little problematic , since there are fallen angels in other religions (Islam) as well.  The names of the specific figures differ.

@@ -11,0 +12,2 @@
+<xref type="see" seq="1056690">サタン</xref>
+<xref type="see" seq="2856037">ルシファー・1</xref>
@@ -13,4 +15 @@
-<gloss>fallen angel</gloss>
-<gloss>Lucifer</gloss>
-<gloss>demon</gloss>
-<gloss>Satan</gloss>
+<gloss>fallen angel (oft. Satan, Lucifer)</gloss>