Insightful, not stupid,
There are some external pumps that are called self-priming.
But these do have limits.
Assuming the pump is not self-priming or placement of pipes and pump aren't within the self-priming pump limits...
...is it just the pump needs to be below the waterline, or does the pump *and* the piping need to be below the waterline?
The answer you want is that the pump
and pipes need to be below the water line. That's a bit general. The pump intake has to be below the waterline but the whole pump doesn't have to be. The main issue is the strainer basket. When the lid is off you want the waterline to be so the pot is full but not below the waterline or else water from the pond will flow out until the pond level goes down enough to be at the top of the pot. Too low a pot can be solved somewhat by adding a shut off valve just before the pot but this is not what most installers would call correct.
The pipes also don't have to be all below waterline. It's just that's the easiest and most reliable way. Technically the pipes can't have air in them. So even if the pipes are all below the waterline if there is a long horizontal run in say a trench and there is a hump in the line then air can collect in that high spot and cause problems (loss of flow and even complete shutdown). Even removing air from these high spots is not always a long term solution because water still contains gases which can escape and the air pocket can happen again.
Running a pipe over the side is this hump/air pocket problem in the most obvious form. The pump is unlikely to achieve prime or it will lose prime over time. There is a solution. What you do is put a tee at the apex of the pipe going over the liner, the high spot. Then generally a reducer to whatever 1/2", 3/4" and a short riser and a ball valve. When the pump is all hooked up and ready to go, the strainer pot is closed and sealed, you open the ball valve and pour water in to fill the pipe. You don't want to hook a garden hose directly because that isn't to code, but you can use the hose to manually fill the pipe. When full you close the ball valve and turn on the pump. It should run for days, weeks, maybe months, until a new air pocket forms and the process has to be repeated. You can also add various valves at different places to make life easier.
However, that's a huge pain and not efficient pumping wise. Running the pipe through the liner is normally how this is done. Today most installers use bulkhead fittings to do this. They're pretty cheap. Long ago bulkhead fittings were really expensive and not always easy to find (pre internet). A guy named Greg Bickal (finally remembered his name) invented the Pipe Boot. You can Google that. It works every well for EPDM liner and just uses a cheap and easy to find pipe clamp. I still use this method.
To reduce risk I send the pipe out of the pond through the side just below the waterline rather than the bottom, even for bottom drains. How far below the waterline depends on where the pipe has to go from there. I like to keep some upward slope on the pipe 1/4" over 4', something like that, so air bubbles will travel up to the pump and generally collect in the strainer basket which are cleared every time the strainer basket is cleaned. Humps in the pipe are not good.
Filter on the bottom of the pond? That doesn't sound easy to clean.