The story is cute

You're right, the translation is quite terrible. Too much mistranslation and at critical places. haha
For example, chapter 3, page 33 to 39, the correct meanings should be:
(Note that this is just my quick translation)
p.33
Incubus: My angel...

ex-angel: ...You're wrong, I wasn't worried. | Nor am I an angel.

Incubus: But you are. (Literal: That's not so.)

p.34
No!!

I...just hate you who won't become mine. | I'm just furious.

Like in the past...

p.35
Back when I was an angel, God was everything to me. I only loved Him. | (Implicit: My love for Him) is purer than anyone's.

But it was not the same for Him.

It's not only me, He loves all.

I became jealous. | Because I wasn't the only one in His eye.

p.36
'Why won't He love only me?'

The moment I thought that, I fell from grace. (Literal: The moment that thought ran through me, I became a human.)

p.37
Nothing has changed since then...

Even now, I continue living holding on to that greed.

Incubus: Such... (Literal: ...That much...)

p.38
Such depths of love you had for Him.

I'd seem that (Implicit: now) it's me who is jealous.

p.39
Because to me,

you are even more important than God Himself.

I think the next one that people probably feel most confused is the last few pages. To be honest I am too lazy to check what the scanlated page-numbers are. So I will just use the tankobon page numbers:
(So starting from the page where they argued on the bridge)
p.176
M: It was at that moment that I met you!

To me, you are the first person who showed me kindness. | The first who held me dear. (Lit: (Implicit: The first person who thought of me as) important.

Is it not enough that I fell in love with you...?

p.177
S: Sorry for misleading you. (Literal: Sorry that I gave you expectation) | But...

It can't be me.

'Masaki' shouldn't fall in love with a person like me a second time. [Hmm...just so that it is easy to understand, S is overlapping the previous M with the new M; since M was given a chance to become human, S wants M to live a happier life.]

p.178
M: Who... | are you talking about...?

Even though I am me... (This line is quite significant because he keeps repeating it. He will say it again at the ending which will make everything click together for S. At this page, the context makes it sounds like he only states the obvious that he is himself.)

p.179

Even though I am not the person who you're thinking about...

p.180
*creak*

SFX from the ticket machine: beep beep

M: Where're you going?

p.181
I knew you would do this. | I'm sorry but I (just) had to follow you.

S: ...Stop.

M: Why?

S: It doesn't matter.

p.182
That...

is the first kiss in my life. (For readers' wishful thinking: it can be thought of as 'human life'. Though as a person, it sort of sounds like 'the first kiss in my life'. I cannot say for sure whether he has regained his memories or not. Because of the '???' and 'I see...' in the last page, I think maybe not.)

p.183
You... | have to take the responsibility.

Since when did you become so forceful/pushy...?
Since I was born.

Because I am me. (At this point it seems that the significance of 'I am me' is one of these 1. The essence of my existence never disappears, with or without the past memories, I am me, or 2. I am the 'me' who you remembered, or 3. both.)

p.184
S: Pff!

Haha!
Hahaha!
???

I see...
I see now... (Here: to me, it seems that he thinks, 'it doesn't matter whether he has the memories of us or not, he is still him in essence.')

p185
Thank you, Masaki.
(This is the first time he calls him with his human's kanji name 真咲--the first kanji means genuine/real, the second kanji means blossom--quite apt actually. Previously after the rebirth of M, the author wrote Masaki with kana when S spoke--because when S first gave him the name 'Masaki' to M, it has no kanji. This line is subbed with kana--this usually happens when the name is first introduced, or well, it could be the explicit way of the author to inform the reader that to S, now they are one and the same person--the author had avoided including the kanji writing of his human name in dialogue from the moment he was reborn you see.
Regardless, one thing is for sure, S has accepted 'Masaki' as a human version of the incubus Masaki; so now to him, they are one and the same.)

With this, I hope you can understand it better. :> Enjoy!

So glad you pointed out that mistranslation 'cause i wouldn't have explained it as well as you did haha. Just to add though that it was probably derived from the story of Lucifer who loved God above all and wanted God to love only him. So Shimizu might actually be Lucifer. This could explain why he still has his memory despite being reborn as a human (cause he's a highrank angel) Really cool if you think about it (๑•ㅂ•)و✧