If standing at sea level and looking over the sea to the horizon, how far can one really see? 13 miles?

It depends on your height above sea level and the height of the object you are looking at. Buster is correct - a person with eye level of about 5' 7'' can see about 3 miles.

An exact answer depends upon the height of the observer and of the object being sighted.

Assuming an observer stood at the water's edge with their feet at water level, a tall person will see a bit further than a short one. Being stood only another metre or two above the water makes a big difference.

Similarly a ship's structure, island or rock protruding 10m above water will be visible from much further away than an object only a few cm above water, such as a floating wooden board or seabird.

While stood on the promenade at Deal (Kent, England), about 5m above the sea, I can clearly see the coast of France, about 40km (24 miles) away. I have stood on the cliff above Giants Causeway, Northern Ireland, and observed the Scottish coast approx 50km (30 miles) away, but I do not know the height of the cliff.

About 3 miles until the water gets in the way. Of course, you can see part of an object further away; how far depends on how tall that object is. The top of the object would have to be close to 100 feet for you to see it at 13 miles.

Yes !!!

However, if you lie on the beach with your eye at ground level the horizon is approximately 2.5 miles away

13 miles is about correct if you are extra tall, and the ship you are looking at is tall.