The word gulf comes from an older word meaning an arch. Find the Gulf of Aden and the Gulf of Siam in your atlas. These are good examples of how the sea bites into the land to form arch‑shaped gulfs. Some gulfs do not form such beautiful arches. The Gulf of Mexico is rather a lop sided arch. Now find the Yellow Sea in Asia. Surely this too is a gulf, even if a bit ragged. And so it is. A sea, it seems can also be a gulf. Though this is not always so.
A sea is a body of salt water smaller than an ocean and in some way connecting with an ocean. The Coral Sea and the Phillippine Sea are right out there in the Pacific Ocean. The Sea of Japan is loosely shut off from the Pacific Ocean by islands. The Mediterranean Sea is surrounded by solid land on all sides. Every gulf and every real sea is connected somehow with the ocean. All the seas and oceans of the earth are connected in some way, The Mediterranean Sea connects with the Atlantic through the Strait of Gibraltar. The Black Sea connects with the Mediterranean through the narrow Dardanelles.
Now take a look at your map of the North Pacific Ocean. Can you see the difference between the Bering Sea and the Gulf of Alaska? Both are bodies of salt water smaller than an ocean and connected with the Pacific, But the Gulf of Alaska is a gulf because much of it is bounded by a graceful arch of coastline.