Should I dig a hole to pump water from under basement?
You should not immediately dig in this spot but you should do some research, get some expert advice and probably should put in a sump system to keep your basement dry.
If you have water entering your basement during rain storms it’s a good idea to implement a drainage system.    There are different approaches but you’re suggesting a sump pit and pump.   I have that in my basement, it’s quite effective.  It’s not necessarily the right solution to put a sump pit in the location you describe, just because you see water there.  Since you have water appearing elsewhere also, you should do some planning to put the sump pit in the best location.
If subsidence is a problem in your area you should not do anything that will remove a lot of water from the soil.   Neighbors and local pros will be able to give you information about that.   If neighbors have sump pumps that’s a good sign.
If  water elsewhere in your basement is coming through the walls you might consider a french drain and sump pit combination.  Read more about those elsewhere, but briefly it’s a trench dug around the inside perimeter of your basement and filled with perforated tubes and gravel to collect water and direct it to the sump pit.   OTOH if water is coming through the floor you might need more than one sump pit.  You’d best get an expert to look at it before you start digging in this one spot.
The location of the sump pit should be where its drain will have easy access to exit the house and then, with as short a run of pipe as possible, drain water onto the ground where it will run away from the house.  You have to think about the outside grade, where the drain will go, and where the water will flow from there.   During a storm you could be pumping hundreds of gallons per hour and you don’t want it all collecting around your house or your neighbor’s.
I don’t think your pool pump experiment will prove very much.   If you don’t have much rain, but you do have a little water in that one spot, and you pump it out …. would that spell “success”?   It really isn’t proving anything, only that the amount of water in that one specific spot given the amount of rain you’ve had during your experiment is (or is not) within the capacity of your pool pump.  It’s not an interesting experiment.
If you have a combined sanitary and storm sewer service, you might be allowed to pump sump water into it but even then it’s not a good idea.  If your main drain line (from your house to the street) becomes clogged during a storm, your sump pump would then cause black water to rise up all through the house.   If you have separate sanitary and storm sewers you must not pump storm water into the sanitary sewer.  Best thing is to drain the pump outside where water flows away from your house.
You should run a new dedicated electric line from your panel for the sump pump. If you have a nearby 20A outlet that isn’t used for very much else you could use that but don’t plug it in to the outlet shared with your washing machine, dehumidifier, or any other large appliance and especially not your basement fridge or freezer.
You should bear in mind that sump pumps need power, and power fails usually during storms, when sump pumps are most needed.  If you rely on the pump to protect a (future) finished basement you should consider backup solutions, either generator power or a backup sump pump.  A backup pump uses no electricity, it uses water pressure from your city water supply to drive the pump during power outages.
Hopefully that helps, but in summary: you should not dig a sump pit in this one spot just because you recently damaged the floor there if you also have water entering your basement elsewhere.  Instead you should plan to dig drains and a pit that will serve your whole basement. You should not run your pool pump experiment because it doesn’t prove anything useful.