Yes those are the common spellings. Gauge is the more common spelling in general public, but for machinists and engineers the shorter gage is commonly used.
From the Starrett website
Choose from a complete range of globally recognized Starrett-Webber gage blocks
Gauge: in my experience is related to something with a dial or has variability. Like a Dial (indicator) gauge, air, liquid or fuel gauge.
Gage: is in relation to something that is a fixed tool, without a dial or adjustment knob, like a gage block or a gage pin, or thread gages.
gauge, tells you how big the bore is of the shotgun.....
And the way you can tell the bore size is, a 12-gauge shotgun, 1 pound of lead will make 12 balls the diameter of the bore. A 20-gauge shotgun, 1 pound of lead will make 20 balls the diameter of the bore. 410 shotgun is the exception, it is like a rifle or pistol, the bore is .410"
As a matter of grammar, gauge denotes something that can have or display a varied measurement. Gage is something with a fixed value or worth since the word is derived from its use as something as a deposit or collateral.
So for the purpose of grammar, anything that has a fixed value the word gage is sort of correct and accepted. For anything that is varied or a result of a measurement with non fixed means, gauge is correct.
