I mean yea, ok, it makes sense inside the battery to point from + to -, but what the physical meaning of being opposite to the current flow (or the E-field inside the wire) is?
For the sake of simplicity let us assume movement of positive charges and forget about electrons. Whatever we state for positive charges can be just reversed in direction to follow the reality. But since all books consider the positive charge convention, let's stick to it. And the sketch I took from a video assumes positive charges.
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There is always a problem with this type of question because in the circuit you have drawn the mobile charge carriers are negative electrons but I think it is easier to answer your sort of question by assuming that the mobile charge carries are positive.
Then to consider the electrons all you do is say that they do the opposite of what positive charge carriers might have done.
Inside the battery a chemical reaction moves positive charges from one terminal, where there is a deficit of positive charges and called the negative terminal, to the other terminal, where there is a surplus of positive charges, and it is called the positive terminal.
That redistribution of charges with the battery produces the electric field pointing from left to right in your diagram.
To move positive charges against that electric field requires work to be done which is done as a result of the chemical reaction within the cell.
Outside the cell the electric field is in a direction from the positive terminal to the negative terminal (the right to left arrows in your diagram) and that electric field drives the positive charges around the circuit with the result that electric potential energy is converted to heat and light in the filament of the bulb.
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$\begingroup$ I think that the circuit which I took from this video youtube.com/watch?v=C7tQJ42nGno assumes positive charges as the current is flowing clockwise in the video. $\endgroup$– bigbossCommented Mar 25, 2021 at 11:58
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$\begingroup$ Also, according to what you state I have 2 things instantly coming in my mind... If the current keeps flowing clockwise it means that in order for it to pass through the source and continue moving in a loop, the Efield pushing the positive charge around the circuit must be greater than the one inside the battery. Let's call $E_s$ the field inside the source and $E_c$ the field that moves charge inside the wire. If $E_c>E_s$ though, that contradicts the video conclusion I linked, that the Poynting vector points outwards the source. $\endgroup$– bigbossCommented Mar 25, 2021 at 12:05
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$\begingroup$ Also you state: Inside the battery a chemical reaction moves positive charges from one terminal, where there is a deficit of positive charges and called the negative terminal, to the other terminal, where there is a surplus of positive charges, and it is called the positive terminal. Actually, shouldn't positive charges flow towards the end where the deficit exists and not the opposite? Did you make a typo? $\endgroup$– bigbossCommented Mar 25, 2021 at 12:11
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$\begingroup$ "Actually, shouldn't positive charges flow towards the end where the deficit exists and not the opposite?" No typo. Inside the battery positive charges move towards the positive terminal. The oppose happens outside the battery where the positive charges move from the positive terminal to the negative terminal.. The gravitational analogue is you pushing a boulder up a hill against the gravitational field and then allowing the boulder to roll down the hill in the direction of the gravitational field. $\endgroup$– FarcherCommented Mar 25, 2021 at 22:57