OK I'll answer your questions one at a time.
Your coolant really needs to be a 1/2 mix of good grade coolant and water, it does make a difference. If you do not have coolant that's available already premixed then simply buy some good coolant and preferable distilled water and mix it yourself. The whole system uses less than 2 quarts so a gallon jug will be fine and where you live I'm sure you use liter instead.
The little plastic tank needs to be empty and just disconnect the hose on it where it hooks up at the radiator cap and if need be use some copressed air to blow it out.
Now since you have used nothing but water I'm sure your system is clean so again take out the drain screw and drain it dry. Again if you have any compressed air you can put a hose in the radiator fill nozzle and add air to blow it all out.
Now I copied and pasted tj burp process and answering another question, yes it's important to let the scooter cool completely. The cooling process allows air to do what it always does rise to the top in this case the top of the radiator fill nozzle where the cap is. Everytime you pull the radiator cap off it you have to add coolant then that means some more air was in the system.
Another thing is to be ready with all of your coolant and caps or organzied. As in the burping instructions take the radiator cap off fill the radiator ful until it will hold no more coolant.
Have the reservoir empty and a funnel in it ready to put coolant in.
Start the scooter leave the radiator and reservoir tank caps off.
As the scooter starts to circulate the coolant most of the time it will go down or recede especially after the thermostat warms up and opens, IMMEDIATELY refill the radiator, place cap on tight.
As soon as you put the cap on the radiator fill the resevoir tank about 1/2 to 3/4 full put cap on and then let her warm up, you can even take a quick spin around the block, watch the temp gage closely and if it starts getting hot shut her down, the fan may or may not come on don't worry about it.
Now be patient and let her as I say get bone cold, remove the radiator cap, check for needed coolant, fill if needed and do it all again.
If you have no other issues like a bad thermostat or failed water pump this will get the air out and make her settle down.
Also I suspect your reservoir tank was too full and it was forcing water out of the top is why you saww water leaking? Other than that if it continues to leak you've got to find out where.
Now here's how I (burp) or get the air out of the system after replacing the coolant;
Now to get started;
(1) Put scooter on center stand and remove carb/radiator plastic cover under the seat.
(2) Place catch pan under the drain bolt location, remove allen head bolt, then remove radiator cap. [DO NOT LOSE COPPER WASHER ON BOLT]
(3) After coolant is drained take water hose and flush radiator well with clean running water, can even do it two or three times. If you have an air compressor you can even blow out all water completely
(4) Replace allen head drain bolt with the copper washer , a little silicone gasket maker is good to use too, secure tightly.
(5) Fill radiator completely with coolant, a long funnel is nice and doesn't make a mess. [USE ONLY COOLANT FREE OF NITRATES, BORATES, SILICATES & SPECIFIED FOR ALUMINUM BLOCK ENGINES] One can mix their own coolant on a 50/50 ratio but it is a lot easier to use a 50/50 premix (I use Prestone)
(6) Start scooter and let run until thermostat opens and you see coolant start to circulate (this only takes a few minutes) Re-fill until full as the coolant will generally go down, immediately put on radiator cap securely after re-filling
(7) Fill radiator plastic reservoir tank to about 3/4 full with same coolant after putting on the radiator cap
(8) Let idle until the temp gauge comes up to about the 1/2 way mark and then shut off scooter (the fan may or may not come on, doesn't matter)
(9) Go watch TV, read a book, but let scooter sit until it's bone cold.
(10) Remove radiator cap and re-fill with coolant, start engine, let circulate, fill again if needed, put radiator cap back on.
(11) Fill plastic reservoir tank again to about 3/4 full, put on rubber cap and watch temp gauge and see if fan cycles on and off.
(12) DONE!
If for some reason it seems to want to get warm and fan not cycle on and off then repeat steps number 9,10 and 11 again.
You can even run the scooter around the block and see if the fan comes on and off and how the temp gauge reads.
This has always worked for me first time and every time.
JR