Weakened draft is causing the leak, If you have a very tight house and have appliances that are using conditioned house air the make up air has to come from somewhere, the stove with the door closed is functioning but as soon as the door is open the house will start trying to find the easiest air to fill the void its experiencing. You can try to trade off the on black 90deg elbow for (2) 45deg elbows and a shorter vertical riser to reduce the turbulence caused by the double 90's the smoke is going through, you can add a piece of class a chimney to the end of your chimney via anchor plate to get extra height.
If you have anything thats using house air, furnace, bathroom vent, dryer, radon air pump, heat pump turn them off and see if that reduces the air pressure issue, also do air sealing upstairs, any open holes say from the top floor into the attic will cause an air pressure issue by taking that upstairs air, venting through the attic and now air has to come in from somewhere else.
I don't think my house is overly tight, it's 1964 construction without permits. That said, I've done some work to make it a lot better (new windows, doors, attic insulation, basement insulation). Also I fixed the attic access so it doesn't leak copious amounts of air, so that's probably out as an issue. The furnace might have contributed to the problem, there is some holes in the return side ducting so it will pull some air from the room the stove is in, but it still can do this with the furnace not running.
I'm conceptually struggling with the door open versus closed issue. The stove does not have an outside air intake kit, initially I was going to buy one (or make it), but the installer cautioned against it as it could cause issues on windy days. So to me it's pulling the same air if the door is open or closed, there's only a difference in volume. I guess writing that out explains it though, there's going to be too much air drawn, lowering the pressure in the room, and that air has to go somewhere so it pressurizes the chimney, if I have that right.
There's no other fans on, air handles, heat pumps, etc.
Is this also partially a temperature thing maybe? It can be extremely cold here, -30 to -40 is normal enough, average days are 5 to 15 degrees Fahrenheit (making me think by converting that).
Is it a basement install?
Yes, in my mechanical room. Which is a plus and minus. The main reason I put it there was the masonry chimney was already in place. But it does a decent job of heating the basement and the upstairs through convection. I've wondered about trying to make it distribute heat better, but that might be another topic.
If the smoke is anywhere near startup
Then preheat with a heater. Make a plywood template for the door and use a small cheap space heater ....The torch is too slow. Try burning kindlin first for 10 minutes or so then add the logs and see if same results.
If the smoke is after 30 minutes in a burn I'd try a wind cap next then switch the pipes.
Changing the pipes does change the draft but this could make your burn times go down...so I wouldn't change the pipes until after you preheat all the time on a cold stove to see if you smoke issues go away....some stoves actually require the door closed or you will get smoke....but cold pipe syndrome is common and easy fix.
The space heater trick will make life better. Get an electronic flue probe and you will see how cold you pipe actually gets and understand it better. It's 16 degrees out and my flue is 55 degrees and I'm not burning today and it's empty stove. So I put space heater on for 5 minutes to get to 80 or 90 degrees in pipe and no smoke leakage.
double wall pipe is a plus when the coals burn down. Keep us posted.
This stove says to not burn for prolonged periods with the door open, which I assumed was because it would get hot enough to damage the secondary air tubes if you did. It may be for smoke issues too.
Preheating it with a space heater seems, umm, like a pain. I could try starting much smaller fires first. Sometimes it works fine, sometimes it doesn't though. I build all my fires top down, I learned it just didn't work building them the "normal" way in my first season. I don't have a thermometer in the pipe, I just go by the one on the top of the stove to give me an idea of how hot the gases must be. I have an IR temp gun, but of course that doesn't work on double wall pipe. So I guess a flu probe might be worthwhile, but I don't really want to spend a bunch of money on something if it doesn't provide me with useful data.
Maybe I should investigate how windy it is when I'm having issues too.