I think it's fair to say that this is an "ancient problem." The ancients only had a compass and straightedge, but they were asking a general question, "can you square the circle?"
They weren't asking "can you square the circle using only these two tools currently known to us?"
The old problem is solvable with a compass, a straightedge and a rope.
[You wrap the rope around the circle and straight it to get a segment of length 2πR, and take the middle point to get a segment of length πR. Then use the compass to continue it with a segment of length R. And then calculate the square root like in https://www.geogebra.org/m/edtecfcv to get a segment of length sqrt(π)R that is the side of your square. I'm sure this was known in ancient times.]
Using only compass and straightedge is more like a esthetics decision.
The old problem is difficult (impossible) because you have strong restrictions about which points you can draw. You have no rope and no magic rule to get any arbitrary length.
The new problem is difficult because you must cut one figure and rearrange the parts to get the other figure.
They have very different restrictions, in spite both are about a circle and a square with the same area.
They weren't asking "can you square the circle using only these two tools currently known to us?"