The answer is no, the two statements do not mean the same thing. And the reason is that they are both incorrect in their details.
Statement 1: "Electrons are a flow of matter and electric charge. Protons are a flow of electric current and matter."
Electrons and protons are not a flow of anything, they are just particles (and they both have mass and charge). If they move we might call that a flow. And charges flowing is called electric current.
Statement 2: "Electric Current/Electric Energy flows from the positive terminal through/by protons to the negative terminal while at the same time electrons with their electric charge flow from the negative terminal to the positive terminal."
- Let's first get the definition clear: A terminal is the name of the end of a battery (so if the circuit contains no battery or some similar source, this makes no sense to mention). The positive terminal is the battery-end with highest electric potential.
Now, try to follow me in the next few lines of thought:
- Electrons as negative charge are gathered at the negative terminal.
- They repel each other, so electrons are squeezed up to the positive terminal. (You can say that they are repelled from the negative or attracted by the positive terminal - either way gives same result).
- These electrons flowing is charge flowing so this is called an electric current.
Protons are not flowing in metal wire circuits, only electrons. In general, as another answer points out, the flow of charge doesn't have to be electrons (or protons), it could also be ions or "holes" and so on. That depends on the type of circuit. So in general, we can simply call it positive charge carriers and negative charge-carriers.
why then is it said that in a DC circuit through a wire, it doesn't matter which way the current is said to flow as long as you are consistent with which ever one you pick in using it in the circuit, if very clearly it is true that electric current and energy start from the positive terminal and flow to the negative.
The answer is: Because people had to agree on something to talk about circuit in general.
Your second statement above was almost correct except the fact that protons do not move, only electrons do (when we are talking about regular usual wire-circuit). This is good to know to understand what happens - but it is not necessary to know if you just wish to work with circuits. Because
- negative charge (electrons) moving from negative to positive terminal corresponds directly to
- positive charge moving the other way.
Even if no positive charge actually moves, this is how it looks from the outside.
So, as mentioned, in some types of circuits (e.g. ion-based), positive (and negative) charge does move, while in others (wire-circuits) it doesn't and only negative charge moves. To avoid these details every time, we just always say that positive charge is moving as well.
And to agree on something, it has simply been chosen that the direction of current is always in the direction of the positive charge. Sometimes positive charge is moving, and sometimes it isn't and we don't care unless we need to look deeper into what charge-carriers we actually have.