Actually the ESC runs the motor.
 
The 450 designation indicates the motor size, a convention used by E-Flite and others.  The 3300 KV rating refers to motor speed, determined by the number of winds, in this case 3,300 rpm per volt.  Some manufacturers use their own motor size designation, such as an A10, which in itself doesn’t necessarily mean anything.

The more meaningful motor specification system is when the physical size and KV are included.  For example a 3542-1000, the “can” is 35 mm in diameter and 42 mm long, and the KV rating is 1,000 rpm per volt.  A 3542-3300 would be the same size but have a KV rating or 3,300 rpm per volt.

I am not sure what a step-less BEC is.  The battery eliminator circuit, may be a linear circuit but more commonly a switching circuit.  Most ESC’s include a BEC, which includes an output voltage regulator, and may supply up to 3 amps of current to the receiver and servos.  The rule of thumb is not to exceed four normal servos, otherwise use a separate BEC, sometimes referred to as a UBEC, with a higher current capacity than 3 amps – and may include a 5V/6V option.  When using separate ESC and BEC, you can use separate batteries for each or one common battery.

Not sure what your question actually is Paul.   Was your grammar teacher an eastern European on a two week visitors’ visa?

Using E-Flite’s data, both their 450 and 480 are limited to 3 lipol cells, one capable of developing 250 watts and the other 275 watts of power.
 
I wonder if the “500” is not of the same motor designation class as the 450, if it can handle up to 6 lipol cells.  Are the 6 cells Nicad or the motor designated to fit a 500 size helo?

This one for example  is listed as 450 class
This motor is specifically designed for the 450 helicopter range and a perfect fit for any of our HK450 helicopters.
Spec.
Model: 2218
Turns: 3.5
Cells: 2~3S
Kv: 3550rpm/V
No load Current: 4.7A/10V
Max power: 500W
Weight: 67g
Dimensions: 27.2 x 36.5mm
Recomended ESC: 2~3S/30A

This one is 450 /500 class

This motor is specifically designed for the 450/500 helicopter range and a perfect fit for any of our HK450/500 helicopters.   with 2 -4 sell  50A ESC
Spec.
Model: 2223
Turns: 4
Cells: 2~3S
Kv: 4400rpm/V
No load Current: 5.2A/10V
Max power: 600W
Weight: 82g
Shaft diameter: 3.17mm
Dimensions: 27.2 x 42mm
Recomended ESC: 2~4S/50A

Paul,

Go for the 500 size heli motor on 6s for your 500 size machine.  3s or 4s will have trouble making enough wattage for a 500.  The physical size of the 500 size is significantly larger then a 450 size and it shows in it's power handling capability (1500w vs 500w).  In the end, it is all about watts (volts x amps).  It is far easier on your electronics to make higher wattages using higher voltages (i.e using 6s for 500 class heli)and lower amp draws.  I believe for a 500 class heli 6s 3000mah packs and 1600kv motors are pretty much the standard.
Hope that helps a bit, let me know if you need any other help with your heli.

BTW, no trouble understanding your question here.

Paul, in addition to what Joe said another important factor is the choice of pinion for the motor. In my misadventures with the HK500 which has a HobbyKing Typhoon motor at 1800 kv I tried to find a pinion that would give the proper head speed on 6S but couldn't find one. I eventually ended up using 5S-4000 batteries on that heli otherwise I had to keep the throttle at around 50% to keep the head speed reasonable.

I can give you more detais about the pinions later if you need it.

Rick