What is the definitively correct way to hook up a 3-prong electrical dryer cord?
There is no way to make a 3-wire connection safe.
It's only legal on a legacy basis, with certain cable types because the appliance industry complained and moaned that they'd lose appliance sales if it was outlawed.
What they are actually doing is connecting chassis to the live neutral.  The concept is that neutral is supposed to be safe... as long as there aren't any loose connections... and loose connections are unlikely since these plugs and wiring are rarely disturbed. Uh huh.
So it is leeeegal to jumper the appliance chassis to the neutral wire and then call it a day.  Would I do it? Heck no!
However in my experience many installations either have a 3-wire receptacle  needlessly because they actually do have ground behind the receptacle... or they are using a cable type that was always illegal (/2 + ground NM or UF).
You are really, really better off upgrading to 4-prong receptacle and plug ASAP.  That will give you a separate neutral and ground.
If your installation is old and uses a legal cable type (/3 noground or SE), then it's legal to retrofit a ground from the socket to anywhere that has a #10 or larger ground back to the panel - water heater, grounding electrode wires, any metal conduit, or back to the panel.
If it is infeasible to retrofit a ground, you can change the receptacle to 4-prong anyway, wire the appliance for 4-prong anyway (meaning: isolate neutral from ground, very important!), and then use a 2-pole GFCI breaker to feed it.  This is labeled &quot;GFCI Protected / No Equipment Ground&quot;.