How does a water heater flue work?

How exactly does the flue in the chimney work? Is there suppose to be a opening at the top, so that steam from water heater can vent out? With the chimney cap it looks like there is no opening at all, for rain or snow to get in from the top

That thing that you call the cap is probably just decorative and is open on the top just like the rest of the flue. There is nothing to stop rain from going right down the flue

Did the chimney guy clog the flue so that heat can't escape

No. If he had, your furnace won't work or you would have been ill or killed by the fumes.

which is causing creosote to leak

You probably need to clean your chimney. Creosote, if that is what it is, is flammable and a fire hazard which can result in a chimney fire

I used to clean chimneys. any stacks that do not have a cap wider than the opening will have the same issue that you have even if cleaned every year as recommended. Some still get water with a cap it is not unheard of with a blowing rain and a fixed cap (if you have one). Fireplaces that are actively used normally don’t see the issue as the heat boils off the moisture , even a draft if the damper is not closed can dry out a bit of moisture.

Looking at the flashing you may be glad this person did not make a cap.

If you examine the tops that ring is there to protect the liner with the regular brick on the outside. Depending on the age of the home. Most with twin stacks as yours looks to have have 2 separate fireplaces or flues for venting at different levels. I would want a cap larger than the opening with wire mesh to keep birds and squirrels out.

I don’t believe that he blocked them it’s just the water washing the cresote down it happens.