Electrical – How should I wire up a 3 prong dryer outlet to 120V

The answer is it depends on the appliance. As long as the appliance does not expressly forbid using it on a 30A circuit you should be fine. You'll be able to change the plug and put on a 14-30P plug, just ignore the silver terminal (neutral).

First, remove that NEMA 10 from the wall with great authority and conviction, and strike it many times with a sledge. Set the resulting fragments on fire. Spread the ashes on the pitch of your town's rival football team while screaming "Demons begone!!"

Convert to NEMA 6

Now here's the illegal part. Get some green tape and wrap it around the white wire, on the insulation near the end. Do this on both ends. (it is illegal to mark colored wires as grounds, with an exception for very large wires).

While you're in the service panel, move the wire from the neutral to the ground bus. (On a MAIN panel, they may be the same thing). You have now redesignated the white wire as ground.

You are now wired for NEMA 6. Put a 6-20 receptacle on the wall and replace the 30A breaker with 20A. Done.

Or, convert to NEMA 14 by adding a ground wire

This is 100% legal. Get some 10 AWG bare or green wire (single conductor, solid or stranded). Run it from the connector site back to either:

• the panel from which this circuit is served, or
• any junction box that has a 10 AWG or larger ground wire running back to that same panel. I.E. your water heater, stove, A/C etc. This has to be a ground wire, not a neutral wire.

If you pull your NEMA 10 off the wall and discover there's already an honest-to-gosh actual ground wire tucked back unused, awesome! Just pop the cover off the service panel and make sure it's actually connected.

Neutral is not ground

All of my advice will fail you if you believe that Neutral and Ground are just two flavors of the same thing. They are not. They serve completely different purposes. Crossing those purposes gets people electrocuted, which is why NEMA 10 is a bad connector. In the 1950s when they wanted to start grounding things, NFPA very foolishly made a compromise with appliance manufacturers, who were afraid appliance sales would go ice-cold if NEMA 14 upgrade was required everytime you bought a new dryer. They compromised at allowing NEMA 10 to continue in service, and use the neutral as ground. The reasoning was an appliance failure (hot-chassis fault electrifying chassis) was more likely than a connector failure (neutral failure guaranteeing chassis is electrified even on a good appliance). Their reasoning was that NEMA 10 circuits serve large appliances which are rarely ever moved or unplugged. Didya catch that part?

Plugging a 15A appliance into a 30A receptacle

You're not allowed to. The issue is overcurrent protection (fusing).

NEMA 6 follows the exact same rules as the old familiar NEMA 5, because the rules do not specify voltage. Any number of NEMA 5-15 receptacles can go on a 15A breaker. Due to a special exception, any number of 5-15 or 5-20 receptacles can go on a 20A breaker. Same rules apply to NEMA 6-15 and 6-20.

A 30A breaker can only feed a 30A receptacle.

So, changing the appliance cord to 30A is right out.

If you must build a 6-30/14-30 to 6-15 cheater cord, it needs to include overcurrent protection (fusing) at 15A or 20A. If you use 20A, you might as well use a 6-20 receptacle, since 6-15 plugs go into it. All the wiring up to the fuse must be 10AWG.

Or, convert the dryer to NEMA 6

You heard right. The trick is, they probably sell this dryer in the Philippines and other east Asian countries that have 240V power. And so they probably make a 240V-only version of this dryer. They designed the dryer from the ground up for this to be possible, and the difference is adding a small step-up/down transformer to power the internal 120V loads.

Don't want to plug/unplug? Sub-panel!

You need a real ground for this.

Remember that bit how those fat 30A connectors are not made for frequent plug/unplug? There's a way around this.

Install a sub-panel at the old dryer plug site. Use one with a main-breaker, any size over 30A will do since it's there to be a switch not a breaker. Don't scrimp on the breaker spaces. 20-30 is not too many. The neutral bus must be separate/isolated from the ground bus, because say it with me, neutral is not ground.

Then install a 30A breaker and bring it out to a 14-30 for the dryer.

Then install a 15A GFCI breaker and bring it out to a 6-15 for the jointer. The 5/6-15/20 receptacles are made for frequent unplug.

This setup has some sizing problems obviously - you can't run dryer and jointer at once for instance. Your local inspector may have an issue with that. But this is certainly the safest of available options. Also the most expense and work.