Pumpkins and squash need to be mature, but you don't usually pick a 'green' or 'tender' squash like you do a cucumber. When those are ripe you should be fine. Tomato seeds need to be fully mature. I know I like to eat my tomatoes while they are still firm, not squishy. The seeds are better to save if I let a few tomatoes remain on the vine as long as possible and I sacrifice them for seed.
Peppers can be tricky for the same reason. Many times I pick a pepper early because of the tenderness, the number of peppers I have available at the same time, etc. Those seeds need time to get mature, too. Since I prefer middle-mild to "so hot it burned the buds off my tongue" I pick those peppers early. Those need to wait for sure to get good viable seeds.
Lots of people have trouble letting their veggie seeds dry completely, but they'll mold in little baggies if they aren't good and dry.
Seeds like chives, carrots, radish have to form on the stalks and be mature, like you said. Plant a few extra carrots you can just leave for seed. You might check to be sure they are heirloom seed or they may be sterile. Lots of commercial seed is sterile (won't produce when planted).
So basically....you were right!