An update on my problem.
The only thing that i didn't do was to use jumcables from a car. because lack of cables and car 
The thing that I did, with an advice from a friend, was to keep the starter button pressed more that 1-2 seconds. And after a few rounds of 7-10 seconds, at some point it STARTED!
This was two weeks ago. Since then, no problem at all, started fine even after a few days of unused.
I still don't know the cause, I will think that was gunk in the fuel.....
Now that the problem was resolved, advice me on how to protect those wires and electrical connections, because I want to clean that area (you see how dirty it is around there 
)
Kymco already has a procedure for this in the manual. It's called the 'anti-flood' because the spark plug can get wet from failed starts.
1) Turn on the ignition
2) Open the throttle
3) Start the bike normally with the throttle open
You might have to try this with the the throttle in different positions until it starts.
https://www.manualslib.com/manual/1636965/Kymco-Downtown-350i.html?page=354
