macbook air early 2008 (original) 13inch: 45 W magsafe charger here
Apple has two slightly different adapters called magsafe L 45W: the first one is recommended for my model, but then on the store they only sell the 2nd one: will this be fully compatible with my laptop?

where (besides amazon) will it be possible to buy just the cord that goes from charger to laptop?
Technically, any Magsafe 1 charger will work. With that said, due to the unique shape of the original MacBook Air, it won't rest flat unless you use the charger that can with it. So you actually don't want any of the chargers shown on that page, you want one like this. Notice the different angle on the charger head. With that said, since Apple doesn't sell official ones any more, be very careful to make sure that you get an authentic one: knock-offs can actually destroy your computer.

To my knowledge, there's no Magsafe 2 to Magsafe 1 adapter (someone else showed a Magsafe 1 to Magsafe 2 adapter which will do nothing for you.) Even if there was, there's no good reason to get one unless you already have Magsafe 1 chargers and then buy a new MacBook with Magsafe 2.