Water – Should I use waterproofing product or caulk at the base of the foundation walls

You have to gather more information about how often the water gets there. Nails without coating (like zinc or anything else) will get rusty if you just wipe them with damp cloth and leave to dry. So nails being rusty only indicate that there was water at some point of time - it might be anything ranging from one time vapor condensation to washing floors to some major periodic leak.

Before going through the waterproofing hassle, do check your gutters again. Even though it's the gable end, you want to make sure your downspouts elsewhere are carrying the water away from the foundation as far as they can.

Then, check the slope of the ground. Is it sufficiently sloping AWAY from the foundation? If not, fix that too.

After that, if you still have issues, if you're going to be digging out the foundation, you definitely want to consider a drain tile system while you're down there. That typically has to go near the footer, so be careful doing that (you'll likely want to consult an expert before digging out the footer.)

At that point, the typical waterproofing system is some form of a rubber membrane then covered with dimple-board. The dimple board protects the membrane from the backfill and also acts as a drainage plane into the drain tile system.