It suggests you have a poor connection at or near the battery. The terminals and connectors should be clean, shiny and tight. Corrosion can creep into the wires themselves at the connectors and cause problems. Inspect, or have someone inspect, the battery cables and the ground connection.

No, you didn't hurt anything by cleaning the battery while connected.

 

I had a similar issue once, and turns out that I burnt out the starter from trying too many times with a low charge battery.
The burnt out starter caused a permanent short/low resistance, so a new battery would either drain really fast or just not even start the car sometimes.

Can't remember exactly the steps that I took, but I took apart the starter and saw that the brushes/contacts were worn out. I either replaced the guts or bought a new starter, but that definitely fixed everything.


Good luck.
I also had AutoZone test out my starter, so you can probably do the same. Although my starter kind of "worked when it wanted to", so might not be 100% helpful.

 

Best way to fig out if alternator is runing ok, start the car, then disconnect the battery from negative terminal ( disconnecting from negative reduce accidental short circuiting) if car stops, means alternator is not producing. If continues to run you can also measure voltage delivered by only alternator from there.
My sister bought her rsx in 2005 from acura dealer used only 1 year by dealership as a showcase car. In first year car failed to start after its parked. It ran ok after jumpstart and failed to start randomly few more times. Went to acura service, battery and alternator came ok. Next day car failed to start again. Second and third trip to service they still say everything ok, they cant find the error or replicate whats happening. When we went to get the car from the service. Service guy was to bring the parked car from the lot. Car didn’t start. They found out there is factory defect in the battery some contact failing time to time when car used , drove on bump etc, some contact disengages inside the battery. if you stop engine that time battery is dead. If you jump start and continue to drive contact may fix itself then everything ok on next start. They changed the battery. It fixed the issue . But they didnt cover it in warranty even car was certified and it was factory defect. Besterds

 

Best way to fig out if alternator is runing ok, start the car, then disconnect the battery from negative terminal ( disconnecting from negative reduce accidental short circuiting) if car stops, means alternator is not producing.

Never disconnect the battery on a running car. This isn't the 1960's. You can introduce a voltage spike that can damage electronic modules.