Your different hardness is likely from a different alloy. Lead doesn't act like steel, it doesn't get harder from a quick water treatment.
And, yes, dropping bullets directly into a bucket of water is a good way to cushion/cool the drop and prevents damage to hot, soft bullets.
Straight wheel weights are quite "hard" to the touch but they don't make real good bullets as they are, they need a bit of tin to help keep the antimony in alloy as they cool. Otherwise the hard antimony and soft lead will seperate and the exposed lead will smear onto your bore, sometimes badly so.
I'll second the comment by yote hunter, that is totally wrong. A re-peat of a bunch of myths, does not make a fact.
Lead alloy can certainly be heat treated. Either by cold water quenching directly out of the mold, or by re-heating in an oven, then dropping directly into cold water. If using the oven method, you should determine at what temp the bullet starts to melt/slump. Then stay about 25 degrees BELOW that to keep the size and form of the bullet stable. Some say there must be some arsenic in the alloy in order for heat treating to work. I dunno if I believe that, most lead-tin-antimony alloys will heat treat just fine without adding arsenic.
The one part of the above is correct, tin should be added to wheel weight alloy, but not to keep the antimony solvent. It will make the alloy melt at a little lower temp, and make the bullet fill out the mold much better.
This would almost be pure lead. A trick old Muzzle loaders would do to check scrap lead to see if it was soft enough for muzzle loaders. As said wheel weights make a great source for mid range loads. With you starting out in casting. I would get either Lyman or RCBS books on cast bullets.
I can scratch a wheel weight bullet with my fingernail. Or should I say any bullet up to around 15 on the BHN scale. I can DEEPLY scratch and even raise a shaving on pure lead. This is shade tree casting at it's best. The only real test is done with a lead hardness tester.