The Shimano freewheel removal tool is not the same as a (current) cassette removal tool. Freewheels take UG (Ultraglide) which is fractionally smaller than HG (Hyperglide) for cassettes. Some aftermarket removers are in between and do either (sometimes it pays to buy cheap!) but the Shimano ones are definitely different.
I sit on the wheel and use a big spanner plus Mr. Mallet.
The minor diameter (at the base of the remover splines) is the same, but the freewheel splines are 0.5 mm tall, whilst lockring splines are 1 mm tall.
Consequently you can use a freewheel tool on a lockring, but not a lockring tool on a freewheel (crappy tolerances aside).
The freewheel tool has longer splines because freewheels get a lot tighter than lockrings, and you need the extra length to avoid stripping the splines (which is why Mr Shimano made the short lockring splines a bit taller).
Be aware that bike shops can and sometimes do describe lockring tools as freewheel tools, so take care. I'd go by the spline length myself.
HG and UG are sprocket tooth designations rather than anything to do with splines as such. UG cassettes had screw-on top gear sprockets rather than lockrings, and freewheels have come in both UG and HG flavours, both using the same remover.
Last time I had to remove a freewheel without the benefit of a good bench vice, it was a 32 mm ring spanner on a loosely bolted on remover, wheel vertical, step up onto the end of the spanner, dip the knee, and kick down hard. It helped that there was a conveniently positioned wall corner next to a door.