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I'm trying to figure that out myself... I guess the left one is the way to go since I believe the right one is making a voltage divider.... but that's a lot of guessing and believing. I'm a newbie in electronics so wait for someone else to answer.
Yes, the right one makes a voltage divider, the left one doesn't. It takes the whole voltage to the gate.
Now, a voltage divider with 10k/200R won't cost you mus voltage, but it is just unnecessary, and sometimes with logic level, voltage to drive the MOSFET is not ample, so every bit helps. If necessary fur your intended driving speed, people report that 100R would still be okish.
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The fact its a voltage divider isn't important, 10k is two orders of magnitude greater than 220.
You can use 100k instead of 10k if you like too.
If the MOSFET was on a satellite board you'd naturally put the pull-down with it so unused channels
behave themselves, but the current limiting resistor depends on the thing driving that board, so
naturally belongs with the Arduino.
The need for current limiting goes up with the gate capacitance - for a small MOSFET it might not
be needed (though there's no way to tell from the datasheet of the ATmega). Once you've talking
big MOSFETs with many nanofarads or tens of nanofarads of capacitance its good to have current
limiting resistor, especially if using PWM.
If you have high voltages in the drain circuit of the MOSFET you probably need MOSFET driver
anyway - the feedback capacitance between drain and gate can overwhelm the gate drive from
an Arduino and put over-voltage spikes on the gate destroying the device. Crss is the relevant
capacitance in the datasheet - say it was 470pF and you switch 100V output in 1us, that's > 10V
spikes with a 220 ohm gate resistor above and beyond what you expect, and the Arduino is
having to push 50mA extra.