To me..
Turn over = Starter turns the engine.
Start = Engine turns and ignition occurs and now the engine is running under it's own combustion power.
Engine Runs = The starter turns the engine and ignition occurs and now the engine is running under it's own power.
For example: The engine turns over but doesn't start = the starter spins the engine but ignition doesn't occur and so the engine isn't running.
kinda.. but there's ON as well. that sometime needs to be explained as well.
to me, turning the ignition ON, is just the first key click. dash lights up. but some people call ON the engine actually running. these are usually new to understanding cars and think of the ignition switch on a car like a light switch in their house. I turned my light switch on but my light didn't come on. they think of cars the same way. not the extra steps that a car needs to start an engine. and then these new electric cars which you do just hit the on button complicated even more. whole they go thru a while sub routine to check for high voltage isolation and stuff. plus those you hit the start button to turn them on, but you can hold the start button without hitting the brakes to put them in the accessory On mode.
anyways.
step 1. key on, one click, eng off.
then key turned to start, starter engages, spinning the engine over, can also be called cranking over the engine.(throw back to hand cranked cars I guess)
engine starts is when you can release the ignition switch from the start position, spring returns it to ON and the engine stays running.