[johnsjmc]

I have seen a few different fresh air intakes setups. Simple-est is a small round duct like a dryer vent without the backflow shut off flap. It comes in the basement wall and turns down terminated just above the floor of the furnace room. cold air flows in to replace the hot air which went up the chimney.
Most complicated is a balanced HRV ( heat recovery ventilator with both in and out).
I,ve also seen a vent directly into the wood burning fireplace box.a grill outside with a screen to keep out mice/bugs and a grill on the floor of the hearth near the front edge. Idea there is supply replacement outside cold air as makeup for the fireplace without it drawing all it,s air from the room.
Looking at your original post I still wonder if you have mortar missing between side by side flues. If the smoke only comes out the clean out when your furnace is running then the idea of neg press in the furnace room could apply. But most high eff. furnaces here have an exhaust and an intake pipe both led outside. If that's the setup then the furnace isn,t causing the neg press. . If someone cheaped out and has the air intake terminated in your furnace room then fix for your fireplace might be as simple as extend the furnace intake outside just like the exhaust.
Also when you light the fireplace the chimney needs to be warm before it can start "convecting"drawing properly.
I had a house which was really air tight after a major program of weatherstipping and caulking. That fireplace would smoke out the front when starting it. Never through the cleanout below that I ever saw.
I began lighting a big piece of newspaper like a torch and held it up the chimney as a first step. It started the draw by partially warming the chimney. Once burning the smoke spillout if any stopped.