koj: キリスト教で、自分の信仰を公に表明すること。
daijs: キリスト教で，自己の信仰を公に言い表すこと。

I really think we're overcomplicating this. The "cultural context" here is universal and I think it can be conveyed with a much simpler gloss.
Added sense.

@@ -20 +20 @@
-<gloss>confession (of guilt, crime, etc.)</gloss>
+<gloss>confession (to a crime, wrongdoing, etc.)</gloss>
@@ -22,2 +21,0 @@
-<gloss>profession</gloss>
-<gloss>declaration</gloss>
@@ -29,2 +27,9 @@
-<gloss>professing one's feelings to someone and asking them to become one's romantic partner (oft. considered a prerequisite to beginning a relationship in earnest)</gloss>
-<gloss>declaring one's love to someone (and asking them out)</gloss>
+<gloss>professing one's feelings (to someone one wants to go out with)</gloss>
+<gloss>declaration of love</gloss>
+</sense>
+<sense>
+<pos>&n;</pos>
+<pos>&vs;</pos>
+<pos>&vt;</pos>
+<field>&Christn;</field>
+<gloss>profession (of faith)</gloss>

I confess to being a bit tired of this one.
I've adjusted the sntences.

Additionally, in 告白① it doesn't matter who 
you're confessing to, but 告白② has to be 
directed towards the specific person one has 
feelings for. It's really quite something 
specific that I think needs to be separate 
from the general "confession" sense.

https://www.tofugu.com/japan/kokuhaku-love-confessing-japan/
KOKUHAKU: JAPAN'S LOVE CONFESSING CULTURE
... kokuhaku 告白こくはく, literally means "confession", and it is done when a man or a woman declares their love to 
another, and hopes to begin dating that person.
... You may go out with the person a few times or go out on a group date, but your relationship hasn't technically 
started until this love confession, aka kokuhaku, occurs.
https://skdesu.com/en/confess-kokuhaku-tsukiatte-kudasai-request-dating/
The art of declaring or confessing love in Japan is known as Kokuhaku [告白] already in the case of inviting a person to 
go out or date after declaring yourself, if you use Tsukiatte kudasai [付き合ってください].
Kokuhaku [告白] literally means “confession”, and it is done when a man or a woman declare your love to another, and 
hopes to start dating that person soon after. The word can also mean recognition and even confessional.

https://ci.nii.ac.jp/naid/40019825078
恋愛における〈告白〉の形成に見る現代日本社会 (unfortunately not available)

https://ir.lib.hiroshima-u.ac.jp/ja/00019550
"恋愛における告白の成功・失敗の規定因"
English title: "Factors influencing success and failure of confessing one's love"
↑
This is a good example why "confessing one's love" doesn't actually work well as a translation. It's not necessarily 
immediately obvious what "success/failure" refers to here with that gloss. How can you fail at confessing your love? If 
somebody told me "I failed at confessing my love" I'd first assume they were saying they just didn't go through with 
it. 告白を失敗

I really think there's a really large cultural context to Japanese 告白 which is something that demands a proper 
explanation and lumping it in with any random confession is doing a disservice to our users who largely are beginner-
level learners. Also this sense is decidedly [vi], which alone is a very good reason to have it separately from the vt 
"confession". 

(kind of irrelevant but there's a very good translation for this sense in Swedish: "fråga chans", except it's only used 
by elementary school kids. https://en.bab.la/dictionary/swedish-english/fråga-chans )

@@ -20 +20 @@
-<gloss>confession (of guilt, love, etc.)</gloss>
+<gloss>confession (of guilt, crime, etc.)</gloss>
@@ -23,0 +24,7 @@
+</sense>
+<sense>
+<pos>&n;</pos>
+<pos>&vs;</pos>
+<pos>&vi;</pos>
+<gloss>professing one's feelings to someone and asking them to become one's romantic partner (oft. considered a prerequisite to beginning a relationship in earnest)</gloss>
+<gloss>declaring one's love to someone (and asking them out)</gloss>