The only real difference I can find is the reference to quantities (but what is a quantity? There are also often remarks about geometry referring to 'real world' objects such as shapes and solids and relations between them.
I'm hesitant to accept these because they fail to recognise the distinction between observations and mathematics.
A slightly more rigorous version of this problem would ask you to find the perimeter of a rectangle with an area of 63 square units and with a length that is seven times the width.
This question statement would force you to assign the variable in addition to solving the problem.
Also, Andrew Wiles' proof of Fermat's last theorem used the tools developed in algebraic geometry.
In the latter part of the twentieth century, researchers have tried to extend the relationship between algebra and geometry to arbitrary noncommutative rings.
So to ask the question more precisely, how do you take a set of axioms and know they describe a 'geometry' or an 'algebra' or any other subject for that matter?
It seems unlikely that mathematicians would label things differently without having a clear distinction between them.
To the best of my understanding, algebra defines You are right, that the attempted distinction makes sense only from a rather naive viewpoint, popular though it may be.
Among professional mathematicians, these inherited "legacy" labels have a surprising endurance, which I think is mostly/only due to their widespread recognition among amateurs and professionals alike, despite their inaccuracy.