However a saucer is attached or located, what matters is that water can drain freely from the pot, and that when the saucer is full it can be easily drained, or the pot is elevated above the saucer so it isn't standing in water and unable to drain completely.
For the first type you describe, often the bottom of the pot is attached near to the top of the saucer, so there is somewhere for the water to go below, instead of around, the pot. For others where the pot extends more deeply into a deep saucer, and the saucer cannot be separated from the pot, you would need to tip the entire assembly -- pot, plants, saucer and all -- to drain a swamped saucer.
I'm trying to visualize how a "saucer inside the pot" would work, but it sounds like there may be a higher "bottom" with a drainage hole inside the pot, so the soil would be held above an area that could catch excess water? If so, that could work as long as the overflow area doesn't become so full that the water level would rise into the potting soil. Unfortunately, it would be hard to tell if the reservoir were full, and impossible to drain without a removable plug at the bottom of the pot!
Keep in mind that far more plants in pots get in trouble from over-watering than under-watering. The potential problem with plastic pots is that their solid sides do not absorb water or allow evaporation as (unglazed) clay pots do. This can be helpful, especially with smaller pots used outdoors in sunny, windy conditions where you might find yourself having to water clay pots multiple times per day. The down side is that plastic pots can retain too much water and get downright muddy near the bottom, especially when the pot holds a new plant with a small root system that does not yet reach throughout the pot.
Because of this tendency to hold water, when using plastic pots it's extra-important to make sure the pot has plenty of drainage holes in the bottom (however the pot and saucer may be configured), and to use a potting mix for that will drain well. Above all, be sure not to over-water plants in plastic pots -- even though the top surface may seem dry, be sure the soil isn't still wet a few inches down.