Since I brought lightning considerations into the discussion let me continue with an example. Consider this illustration taken from the IEEE document I referenced.

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It's a common example of a central A/C installation. Note there are two grounds one intended and one not intended. The not intended ground may be because it's a metal cabinet condensing unit or the uninsulated copper refrigeration lines touching wet earth. The intended ground is the earth ground at the service entrance. Now when a lighting strike occurs, induced currents are really out of this world huge. 30,000 amps in the example. The earth becomes a conductor between the two grounds, the intended service ground and the not intended ground. 25 ohms in the example. V = IR ; V = 25x30,000 = 750,000 volts! Poof went your A/C motor insulation! This example shows why you only want one earth connection in your house wiring and outbuildings. You don't need to have a direct strike. A nearby strike to a tree or other object will still induce ground currents in the path of your two or more earth grounds.
How to protect your equipment? Several ways, be sure your wiring is correct. Identify all the possible conductors coming into your house, electric lines, telephone wire, CB antenna, metal water pipes, metal sewer lines, etc. Think through what could happen when lightning induces a current between any of them. If electronic items are included in the current path between two or more of the grounds it will be damaged. If there is only one earth ground there will be no damage because the absolute voltage will rise but all parts of the system will rise by the same voltage and there will be no current produced.
Use a whole house surge protector. Here's an example of a Square D brand in my service entrance panel. The protector needs to installed correctly, follow the instructions for wire trimming carefully. During a surge it's really a high frequency transmission line.

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Also use local surge protector at all electronics. These work because the lightning is really a very fast rising pulse that has difficulty traveling on house wiring. The inductance of the house wiring attenuates it. By the time it gets to your local surge protector it's reduced enough that the varistor in the surge protector can absorb it.
My house uses the metal well casing as the earth ground connection. When my house was built all well casings were steel pipe. Now plastic pipe is used. I've been told that well pumps burn out more frequently when you have a plastic well casing. Think about it. That pump motor is a dam good ground, it's in the water. Not surrounded by metal and poof. Read the referenced document for other examples. There are other references at the end for more in depth study.