The designation of what is "high" and what is "low" is actually done by convention. Nature actually doesn't care what we call it. So assigning something to be + or - is also by convention (or in this case, tradition).
negative pole is of lower potential and +ve pole is of higher potential and it is taken that more electrons in the negative pole due to their negative charge.
The electrons do move from the negative terminal to the positive terminal,
both statements are coreect and i think its correct explanation
P.S: Any kind of answers and comments from other experts are most appreciated
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regards
Ramesh