Bad Cold Weather "Flue" Season! : (

"air from the outside is pushing the fan blades in a fairly strong manner" means the building is depressurized. I'm guessing there is a kitchen nearby with either defective or insufficient MUA.

You need to find what is depressurizing the space and fix it. If not, isolate the heater area and install an external makeup air to this space only. If you go down this route make sure the verify the pressure in the space when done.

Since hot air raises, a chimney can't draft down into a heated space. The only way for cold air to come down through a chimney (assuming chimney is above roofline) is for the building space to "suck" the air in.

Sometimes if you have a place with a lot of air leaks near the peak, stack pressure will create enough airflow to pull the basement negative. Solution is still the same, stop the negative pressure source or add enough external makeup air to prevent back drafting.

The other issue could be if the chimney is exposed to the outdoors. A cold chimney can backdraft. From the picture with the cap, looks like inside the building, so I don't think this is the issue.

Are there any large power vents on the roof? Air handler on the roof or inside unconditioned attic? Look for anything that can suck the building negative.

The solution is still the same, more makeup air. Once you add enough that the chimney isn't the source for makeup air, it will stop back drafting.

@RickDelta OK. This is obvious that there is a lower pressure inside the building, and/or there is a higher pressure outside the building. You can come up with some way to identify the source of the pressure difference and then mediate it. This may not be easy to do, and even if you did find it , you can't just. turn it off. Someone put it there for a reason.

I was in a similar situation in Philadelphia when a local Diner called me to solve a problem with a water heater with a standing pilot that kept blowing out. The water heater was in the same room as the heating boiler and vented into a 12x12 terracotta lined masonry chimney. There was a metal door on the boiler room which had a louver on the bottom of it. It seems strange to have a metal door (like a fire door) in a boiler room with a louver cut into it for combustion air. There was a glass block basement window in the room also. Now that I think back on this job from over 40 years ago I guess that was a fire door when the construction was inspected many years earlier and that window was blocked up to stop unwanted intruders from entering the building through the functioning basement window that most likely had combustion air louvers. But that was not my concern at the time, I just wanted to figure out why there was such a strong downdraft from the chimney.  That room communicated with the rest of the building thru that louver and again thru the louver in the basement door up into the kitchen where there was a huge exhaust fan over top of the main cooking station oven/grill area.

No way I’m going to get a natural draft thru that chimney with all that exhaust power in the kitchen. You could actually feel the pressure at the front door of the  restaurant when you tried to open that door to come in.  The suction actually made it difficult to open the door until you got past the first 6”. Then the door opened rather easily after that.  

So I went to the supply house and purchased the most powerful draft inducer they had in stock and mounted it on the common vent pipe of both exhaust connector pipes connected to that chimney.   Never had a problem after that.   With all the other equipment that was involved with the building, adding that fan didn't amount ot anything as far as power usage so I didn't even wire in a switching kit for the draft inducer… I just left is on 24/7/365. there was always air moving up that chimney for the water heater to vent properly. And I guess the front door was just a little harder to open after that.

Get yourself a larger draft inducer and get rid to that goofy chimney cap. That was a waste of $$$, time ,and effort.

Draft inducer DI-4

If you do not want to operate the vent motor 24/7/360 then you need to get this

Control kit

Now many will say thet this is not the best solution, that the DI-4 will place the masonry chimney into a positive pressure and that the terracotta liner is not supposed to be under positive pressure, and they are correct. the correct venting system would be a chimney topper draft inducer like this one. https://www.fieldcontrols.com/type-c-chimney-top-draft-inducers/#1680177330201-ecb61e8a-511c

But that one is much more expensive and may not be large enough for your 16" pipe and 18x18 tile liner. But you can see if either one will solve your problem.

You need the vent on the outside on top of the chimney as you want the flue at negative pressure. A blower at the cleanout can pressurize the flue and create more back drafting issues.

@EdTheHeaterMan solution is the proper one in this case as you won't be able to eliminate all the negative pressure sources in the building. Even if you reduce the bath fans, you still have the exhaust fan and the dryers in the laundry room.

Getting balanced ventilation or at least reducing the amount of exhaust would definitely save some real heating cost for the owner.

Here is how the air will move with a fan placed in the clean-out opening at the floor

Red areas indicate a higher pressure zone

Blue area indicates a low pressure zone

White areas indicate places that will let the pressure can equalize; like windows, doors, chimneys and fresh air intakes.

If you look at the boiler room, the blower fan will cause the boiler room to have a lower pressure than the rest of the building. That blower will cause the outside pressure to be a greater difference making the chimney down draft even greater. The blower fan can push air up the chimney to the vent connector location, the outside air pressure can downdraft to the vent connector location and there will be no exhaust gas going up the chimney, so all the exhaust gas will accumulate in the building. 

The rule is that high pressure moves to low pressure,