Is having yellow watery discharge a sign of wound infection?

Yes. Signs of wound infection include redness, increased pain, fever, red streaks on an extremity, a wound opening up, or drainage of fluid or pus.
Drainage normal. Anytime you have an open wound, inflammation results. Swelling (edema) in the tissues leaks out when there is an open wound. This does not necessarily mean infection as it is a normal physiologic process as is the redness, warmth and tenderness. Knee jerk antibiotics are a bad idea as they can cause superinfcetion and allergic reactions with no benefit. Antibiotics are always the last resort.
Yes. The signs of wound infection are redness, tenderness, swelling and fever. Purulent drainage almost always indicates a wound infection.