I don’t see why the ignition will draw any power with the key in ACC position. Yes, if you turn it to ON, then it will power ignition. You can easily turn off the interior lights in all my cars to prevent them from illuminating unnecessarily but when I do vacuum, it only takes me 10 minutes so no big deal either way.
Sure, pull it out, take it to an auto parts store with a tester and have them check condition. That will give you the best test.
You could also get a digital volt meter and check the battery voltage with the car off, after you’ve driven at least half an hour and let the battery sit at least half an hour. The voltage should be at least 12.7 or more volts. If it is less than 12.3 volts, it’s near the end of its life. Not a perfect test as it does not test the load the battery is capable of delivering but its a quick test with reasonable accuracy as long as the battery still successfully starts the car.
A tip to the OP: turn off your interior lights while you are cleaning the interior for an hour (unless you are doing the work in a dark place).
The lights will drain more from your battery and shorten the lives of the bulbs.
You can measure the specific gravity of the electrolyte with a cheap device from the auto store.
I switched to an LED light that takes little power and is far brighter. The domestic suppliers were expensive ($12); an equivalent from China on eBay cost about $1 and has worked well. I’ve spent hours parked at night when I wanted to read; the LED light was bright enough (the standard bulb was nowhere near bright enough.) and used less power.
…or you could leave it in and just drive it to an auto parts store. They don’t need to remove a battery to test it.
…and isn’t this a moot point anyway? The device says on the warning label to run the engine while you use it and the OP said the vacuum cleaner runs better when the engine is running. Why are we getting caught up in these trifling details about whether it will run down the battery?
The original question was “…should I only turn the car on, or should I turn the key all the way to actually start the engine?” The answer is “The device is telling you to run the engine, in more than one way.”
The best way is to ask your mechanic to do a battery load test. They have special purpose equipment to do this accurately and easily. As posted above some auto parts stores offer this service gratis for their customers.
A diy’er method if you have the skills & necessary equipment (inexpensive volt meter) is to pop the hood & measure the battery voltage after the car has been sitting overnight. It should measure around 12.6 volts at 70 degrees. Make sure everything in the car turned off, doors closed, etc. Now turn on the headlights to bright. The voltage shouldn’t drop more than 0.5 volts. The more it drops, the worse condition the battery is in.