It’s a complicated process (the chimney draft) and since I know nothing about the design of your home I cannot give you any definite answers.
Proper draft depends on many conditions inside and outside of your home; chimney type (material), its dimensions, height above the roof, positioning (inside the house or outside), number and type of appliances using the same chimney, tightness of your doors and windows, additional vents and chimneys inside your home (fireplace, exhaust fans, etc.), weather conditions / temperature / barometric pressure at the time of testing, altitude, and a bunch of other things.
A heating assessment company should check all of the above, input it into their software (or whatever they’re using) and based on that information produce some answers. By analyzing it they should be able to tell if the behavior of your chimney is normal or not.
In some cases a water heater chimney will have no draft at all until it is stimulated somehow, by a water heater’s burner for example. As soon as the air temperature inside the vent connector and the flue starts rising, created differential between inside and outside / ambient temperature helps to pull exhaust gasses outside… that’s extremely simplified version of it.
None or poor draft may happen in an extremely air tight home where there is a problem with balancing fresh air intake (from exterior), or in a home where there are too many constantly active exhausts that create negative pressure and downdraft inside the chimney.
Since you have to start somewhere you may want to call a local chimney sweep, maybe there’s something obstructing the flue but your CO alarm would most likely alert you about it.
Let me know if you have more questions and when you find out what’s causing the draft problem (if it is a problem).