How should I connect a separated 5 V source to an Arduino connected to a RAMPS board
I would not recommend changing Arduino powering.
You can either do as you say and power Arduino by giving it regulated 5V to Vin, but then the ATMEGA2560 still will be powered through the 5V voltage regulator. The voltage regulator (NCP1117ST50T3G) needs to be supplied with 6.5V to give out regulated 5V. So you will be powering the Arduino with some 4V, which might work, but it will be less reliable. But it is electrically safe.
Second option is to power Arduino through it's 5V pin. That is less safe, because the voltage regulator might have an over output-over-voltage-protection that will to load or short-circuit the 5V until it is 5.000V.
Regarding TMC2209. I have not seen any documentation that has any opinion on what order it shall be powered up and down.