That device will certainly tell you if your battery is bad, but I doubt that is your problem. If it were a weak battery, it would NEVER start, regardless of how many times you activated the key. Activating the key won't somehow make the battery stronger.
Make or borrow a test light. A 12 volt tail light socket and bulb from an old vehicle works great. So does a sealed-beam headlight. Just attach some leads, each a couple feet long. Alligator clips make it handy.
Hook the test light to the small wire on the starter or the starter solenoid. Activate the key and see if power makes it that far.
If so, attach the test or to the BIG wire at the starter. Activate the key. This will tell you if power is being made available to the starter motor.
Hook it to the battery cable right at the battery post. If it lights, great. If the light goes out when you hit the key, that's a poor connection right at the battery terminal. (if the tester lights up, the connection is good enough to supply a couple amps to light up a test light, but if it goes dim or completely out when you demand several hundred amps at the starter, that's a poor connection)
Test the negative side of the system as well. Power not only has to leave the battery and make it to the device, but it also MUST return to the battery via the ground (negative) terminal.
If I were to offer a WAG, I'd be betting on a bad connection in one of the big battery cables, either at the starter or at the battery posts.
If sourcing a test light is problematic, disassemble each big battery cable connection at each end, wire brush them (and the base metal, in the case of the ground cable) and reassemble. Don't forget each end of the NEGATIVE cable, as well.
A redneck test is to just HOLD the key in the "click" position for several (up to 10-15) seconds, then go around around and feel for a hot connection or watch for smoke. A bad connection will produce heat when you demand a load thru it.