Well it depends on what kind of positions you like playing so it's hard to add anything blind without further info. However, based on 2 out of your 3 displayed choices it looks to me you enjoy attacking and initiative so I'd suggest you take a look at evans gambit after you try to go for fried liver and black replies with 3..Bc5 instead of 3..Nf6. Really fun to play and lots of attacking potential as opposed to the more solid 4 knights italian (although keep in mind 4 nights italian is technically better than evans gambit)
Still on the 1.e4 e5 topic... after you play Nf6 people somtimes play Petroff (2..Nf6) and philidor defense (2..d6) instead of the more common 2..Nc6 to defend the e5 pawn.
In the Petroff defense you can either play the main line by capturing e5 pawn with your knight or develop your bishop to c4 - Urusov gambit - gambiting the pawn on e4 and then it can transpose to fried liver if they decline the gambited pawn and play Nc6... but they can also capture the e4 pawn or play Bc5 and that would be too much theory to cover here but they are all very playable.
If they play the somewhat common philidor defense, then the best reply is d4 and if they play Bg5 pinning your knight I recommend you look into Morphy's Opera game as you'll see some cool ideias. If they capture the pawn exd4 then you can recapture with your knight or gambit another pawn smith-morra style. Maybe even 2 pawns danish gambit style.
Now.... if they play f6 to defend the e5 pawn, aka Damiano's defense you can capture the e5 pawn by sacking the knight on e5. Then they have to play Qe7. If they recapture with the f pawn it's a winning position for white, which is cool to play and happens somewhat often below 1000. Youtube how to crush Damiano's defense or something and there should be some interesting material on that.
Besides that the most frequent replies you'll see against e4, besides e5 and c5 sicilian, are scandinavian, french and caro-kann. A little less often you might run into modern defense, pirc defense and owen defense. And an occasional alekhine's defense. But tbh that is a lot of theory to memorize (as you can see it's a lot of info here on this post for 2 moves of a couple of lines on e5) because of many different variants that can happen especially cuz they are not played that frequently and when they are played it can be a main line or a side line or pretty much anything and unless it's an obvious blunder it's hard for people our level to punish inaccuracies. So just stick to opening principles against those and you should be fine.
All of that being said, don't spend too much time by memorizing lines. I see that you analyze your games after and that's great. If you end up memorizing by osmosis by analyzing it, fantastic, but don't spend too much of your time just looking into theory to try and memorize and hope it happens in a game. Best you can do to get better and win games is to keep it simple. Develop your pieces, control the center, get your king to safety. Pretty much if you don't hang anything your opponent will. And if you know how to trade down when you have material advantage, then promote your pawns to easily checkmate your opponent you will get to 1000 rapid real quick. Since you have a premium account do some puzzles to be able to spot tactics in your games and you'll find yourself having a lot more success than by simply memorizing lines.
Good luck hopefully this wasn't all too confusing
PS: If you play the fried liver watch out for the traxler counter attack. If you haven't fallen for it you might. It's not a good move for black but you have to know how to reply to keep your winning position as white
I've been strictly playing e4, and this is my repertoire so far:
- Fried Liver attack
- 4 Knights Italian
- Smith-Morra gambit (against Sicilian players)
What are other common openings you guys run into when playing as white? (As black I know a little bit about every opening and can usually get by just fine. Ideal scenario as black for me is a white player trying to Fried liver me, as I know a ton of theory for the fried liver attack as black and as white.)