What is the name of this system in the English Opening?

Here is the basic system for white (ignore black's moves):

Does this system have a name? I just want to find games to study in this system

Yes, this is the Botvinnik System, I've used it to decent success before. Sometimes in bullet/blitz and almost always as a surprise weapon/fun  White is mostly playing for f4!? in many lines. I don't use this in my default repertoire because Black has several good equalizing lines against it - however any system is solid under short time controls - especially if it offers attacking potential like this one does.

https://www.chess.com/openings/English-Opening-Botvinnik-System 

In your diagram, White has yet to play d3 (and has played e3 instead of e4) but this looks like the opening system you are after. e4 looks funny because it blocks the fianchetto Bishop but it works in this line. d3 is also fine, but even in d3 lines, d3-d4 is often times prepared in the middlegame.

Incorrect!  This is NOT the Botvinnik System, and the Botvinnik System would be HORRIBLE here with the Black Bishop on c5.

The Botvinnik System sees White placing the pawns on c4, d3, and e4, NOT e3.

There is no name for White's structure here.

The what should we call it? Even if it doesn't have a formal name, we should give it something for identification. 

At least to me, I figured it was just a bad version of the Botvinnik System (because of the differences I noted). If this isn't even a bad version of the Botvinnik, then what should we call this setup?

@Oakus is the first I know to come up with this, so maybe the "Oakus System"? Sounds as good as any other opening name to me.  Now all we need is some GM to play this line once or twice to popularize it and then "steal" the name lol

I think it's some sort of a combination between the e3 and g3 lines, I don't know what it's called ...

Agreed. It is similar to these, but it is not named (as @ThrillerFan correctly states). I was thinking maybe this was just an inferior Botvinnik setup, but it seems to be different enough to get its own opening name

U could say its called "korchnoi" system coz korchnoi used to play it, but alexander morozevich introduced the f4 idea as far as I know

The system does not have a name. There are several variations, where White uses this set up or something very similar. For instance this is a line in the Wedberg variation (where the name refers to the black set up, White can also develop the knight to f3, an example is here: https://www.chessgames.com/perl/chessgame?gid=1633757), one of the main lines of the reversed Dragon differs by just inserting ... d5 cxd5 Nxd5 and there are lines in the reversed closed Sicilian, where White uses this set up.

The Botvinnik system is very different as there the pawn system is very solid but it also blocks White's own bishop and pushing d4 is very difficult.

No because the botvinnik is specifically c4-d3-e4 with a gaping hole on d4.  Calling the e3 lines the Botvinnik is like calling the Classical Dutch the Stonewall (or saying it is a bad Stonewall).

You cannot even call it a bad Botvinnik as d4 is covered and the a7-g1 diagonal is blocked.

Actually, due to where the Black Bishop is, c4-d3-e4 would literally be a Bad Botvinnik.

The Botvinnik only works against 1...e5 lines where Black fianchettos Kingside, or against Dutch lines (1.c4 f5).

Fair enough. I think part of it is just me trying to force a name upon this setup. Maybe "Korchnoi System" like @vishnu_vijay_93 came up with

Not every system has a name.  It probably would be in a chess player's interest to not have any names.  That would force you to understand moves naturally.  Saying that e3 instead of e4 is a Bad Botvinnik shows that one does not understand the Botvinnik at all.  Going one square instead of 2 changes the entire perspective of the position, and going 1 square is not bad.  If going 1 square instead of 2 constituted a bad version of another opening, you'd have:

The Classical Dutch is a bad Stonewall

The French is a bad double king pawn opening

The Caro-Kann is a bad Sicilian

The Pirc is a bad Scandinavian

The Triangle Defense is a bad Tarrasch Defense

This is all nonsense!  The Botvinnik System has a specific point.  To keep a firm grip on d5, even if it means a major weakness on d4.  The move e3 does not control d5, but it also does not severely weaken both d4 and the g1-a7 diagonal.

Trying to force names into every variation just deters where your focus needs to be if you are looking to improve at all.  Take the English Opening.  You do need to know what the English Opening is.  You are playing 1.c4 and you ask about the name of that.  That is necessary to know what subject you need material on.  The English, The French, and The Spanish are three VERY DIFFERENT openings.

But once you find out that you need to study the English, you should not be so engrossed on names of minor variations.  Yes, it would be good to know names like the Symmetrical Variation (since there are entire books on just the Symmetrical) or the Reversed Sicilian.  But after 1.c4 e5, you need to focus on understanding, not memorizing names or moves.

What are the pros and cons of 2.g3 vs 2.Nc3?

If 2.Nc3, what are the major differences between 2...Nc6 and 2...Nf6?

What are the major differences in the development of Black's Dark-Squared Bishop?  How is going to b4 different from going to c5; and how is going to b4 or c5 different than going to g7?

What are the major differences between White playing an early e3, an early e4, and an early g3?

Which pieces should White be looking to trade and for what?  N for Light-Squared Bishop?  N for Dark-Squared Bishop?  N for N?  B for B?  Either Bishop for N?

This is where your focus needs to be, not sub-variarion names.

Valeri Lilov showed this structure in one of her videos on the English. She said it was 'a bit simpler to play than the Botvinnik System' - but she didn't say what it was called. I found it quite appealing for this reason, but I too was puzzled as to what it was called. I searched. And I searched.. Until I spotted the 'Taimanov variation on the English Opening'. It does bear a certain similarity to the (old) Sicilian Taimanov (with colours reversed), where black has a knight on e7 instead of the way the Sicilian Taimanov is now played with a knight on f6. 

The only reason I wanted to know the name of this system is to find and study games played in it. Is there a way I can find these games without the system having a name?

Sure. Lots of opening databases are out there; simply "play" the moves on the board and it narrows the search automatically. However, both White and Black moves are specific; this means that you can't just "ignore" the moves for one color. Some databases also have an option for counting transpositions or not including them.

In addition to the chess.com Opening Explorer (which may require certain membership levels to access), here is a simple Opening database that is free to use:

https://www.ficsgames.org/openings.html  

Open up chessbase or whatever chess program you have, like Fritz includes a database but it is smaller, and set the position on the board, like after White's 5th move, and then search for position.

Daniel Naroditsky commonly uses this setup against closed Sicilians. He calls it the Botvinnik, but the "gist" of this configuration is so incomparably different from the e5-d6-c5 setup that it just needs a name of it's own.

It's definitely not worse than the Botvinnik with e5-d6-c5

I think this is the Korchnoi system

I don't think this is a system that can be comfortably played regardless of black's moves. It looks to me like a reasonable line against black's play as presented here. Back when I was first learning to play the English I would play something like this against almost anything and occasionally get absolutely killed on the light squares. This really only works because d4 is going to immediately hit the bishop on c5.

This is a Closed Reversed Sicilian English