Weather-stripping can be in compression or in friction. Compression sometimes messes with the latch mechanism. Also, if the 2 doors aren't in perfect alignment from top to bottom. Compression may leave a gap at the top of bottom.
The piece that is normally between double exterior doors is known as an astrigal, as mentioned. The bulb type in drawing #1 is a little bit of both. W/o a kerf for the tab attached to many varieties of bulbs it's hard to glue and keep in place. Too much force over and over as the doors operate.
The best solution is to install a complete new astrigal that incorporates w/s. Finding one in all wood is difficult, but extruded AL models can be picked up at the box store.
These off a lot of security as well as helping stabilize the alignment. Generally you start with aligned doors with a 1/2" gap between the 2 leading edges. So, yes, if you're not much of a carpenter, you might call one. About a 2 hour job. I've installed literally hundreds of these.
Here is a simplified version that I don't favor but they work if everything is in alignment. I know I used that word quite a bit but that's the key to this problem and solution.
