I am fine when I see Spanish, but absolutely freak out when someone plays 3. Bc4. Looks like it is just me here with this syndrome. Bologan's book has both Bc5 and Nf6 variations, but Nf6 seem to have less variations. I am yet to reach the main lines chapter, pondering over the sidelines currently. Let us see what happens (although it might take a month or two before I complete the chapters and start playing again). Enough is enough. I just do not want to fear or loose to Italian ever again.
Dealing with the Italian Game as Black

Reminds me how few months ago I was getting beaten convincingly in the Spanish exchange variation. I just didn't knew how to handle it. Finally I came across a Be6 sideline (f6, Bc5, Qe7, 0-0-0 setup) and I have never been outplayed again. I have lost games, but not outplayed and pushed into passive positions since. Now I pray for BxNc6 everytime.

To illustrate my fear of Italian Game and how badly I play it, take a look at this game. I just can't seem to even play out the first few moves properly, and blunder. My opponent (claims to be FIDE 2100) is convinced that I am too bad, turns greedy and goes for pawn grabbing which helped me setup an attack against his King. I was totally lost had he played sensibly. I am this bad with the Italian Game, and I am going to hopefully turn it around in the coming months.

To illustrate my fear of Italian Game and how badly I play it, take a look at this game. I just can't seem to even play out the first few moves properly, and blunder.
But you played the Hungarian defence. It is really passive. The game you have posted is not typical of this opening.
As you said earlier, 3. ... Be7 will work against some weaker players. If you are playing someone that is certain to hang a piece or allow an elementary fork or skewer, before you make a mistake yourself, then you can get away with openings like this. Against stronger players you need activity, or you'll be gradually suffocated.

Yes, I just wanted to show how I blundered a pawn at move 5. I have tremendous dislike for the Italian overall since years, and hopefully learning some lines will ease the psychological thought process as well.

That is because the mainline goes with dxe5 not Nxe5.
The Hungarian is not a bad choice. And even tho databases show really amazing results for White after dxe5 dxe5 Qxd8+ Bxd8, I find it hard to believe White is that much better.

Frankly it was the first time I played Nxe5 thinking "how bad can it be? a minor piece will be off board soon" until I saw his Queen visit h5 with double attack.

That is because the mainline goes with dxe5 not Nxe5.
The Hungarian is not a bad choice. And even tho databases show really amazing results for White after dxe5 dxe5 Qxd8+ Bxd8, I find it hard to believe White is that much better.
I am sure that objectively you are correct, but that does not make the Hungarian a good practical choice, especially if someone is playing it without having made a special study of it.

https://www.chess.com/blog/jengaias/italian-for-black part 1-5.
Take a look , it will give you a starting point.
Awesome! Thank you. I will go through it all.

The Hungarian is just an endgame, not study required.
Surely anyone playing this opening ought to have investigated the early endgame to ehihc it can lead?

You know what I mean. Yes, probably the OP will have much better results if he familiarizes himself with the endgame, but I think you can easily play the endgame without knowing it much, it is just a chess position.
I would much rather preffer to be out of book in that endgame that out of book in a Najdorf
Back in the days when I routinely answered 1. e4 with 1. ... e5 I was much happier to see 2. Nf3 Nc6 3. Bc4 than 2. Nf3 Nc6 3. Bb5. It is much easier for Black to play against the Italian game than against the "Spanish Torture".
If you do not want to spend a lot of time on the main lines then it is easy to master the variations where Black maintains his pawn at e5 with moves like Qe7, Bb6 and d6 (exact order can vary). Very often White castles early and Black can have all the fun with a K-side pawn storm. The only other extra preparation you need make is some system against the Evans gambit.
Someone also suggested the Two Knights. That has the big advantage that it will take many opponents out of their comfort zones, but the disadvantage that there is a lot to study if you are to avoid the many okay looking moves that are just bad. It is not the sort of opening that you can make up at the board, unless you are some rare genius.