Oil pressure light periodically comes on

1. As a basic measure I would make sure the correct grade and spec oil is going in the engine. You can no longer trust that a local mechanic or even a dealer service center will necessarily use the required product as stipulated by the manufacturer. It is always better to buy and supply your own so you know the correct product is being used.

2. I would not assume there is a problem with the oil pressure warning light even though there could be.
The first thing the mechanic should have done when you reported the oil pressure issue is to hook up an independent mechanical oil pressure gauge and check oil pressure when the engine is hot at different RPMs. Any oil pressure issue will usually show up at idle when the engine has been warmed up properly and the engine oil is hot (unless you have some kind of intermittent oil pressure relief valve that is sticking). Just replacing parts without actually knowing what the oil pressures are at different RPMs is being sloppy and negligent.
This is what you need to do now. Contact a business that can hook up their own pressure gauge to your engine and have them check the oil pressure readings at various RPMs when the engine is fully warm. If there is a low oil pressure issue it will show up on the oil pressure gauge. As you have a new oil pump there should be no issue now with oil volume or a sticking oil pressure relief valve which on your engine should be part of the oil pump assembly.

3. Worn engines will not be able to sustain required oil pressures. Oil pressure is not produced by the oil pump. The oil pump produces oil flow but the resistance to flow is what produces the oil pressure and that resistance is created by fine clearances between operating engine parts especially on the crankshaft and components connected to it. Once the engine wears the clearances increase and the oil flows past the parts much more easily and produces/builds much less pressure.
How well your engine with over 90000 miles has been maintained depends on how often all the owners changed the engine oil and filter , whether they kept the engine oil level up to full (most owners rarely ever check oil levels between services) and whether they always used the correct product. Manufacturer oil change intervals shown in service schedules can be 10000, 12000 miles or more which is much too infrequent for the type of use 95% of private vehicles experience which typically are on the severe to very severe use end of the spectrum. Engine oil typically needs to be changed at least at half the mileage shown in service books (at least twice as frequently) and thus at least every 5000 miles if you want your engine to last.

have the oil pressure check mechanically to remove the pump as a source of the problem
check the oil level in the sump to remove that as a source of the problem
replace the oil pressure sender or sensor unit

May need a new oil pressure switch,it's located next to oil filter,remove electrical connector then part.You may have to remove oil filter to remove part.

It's OK that oil light flickers or comes on while starting. It turns off when oil pressure comes up to normal. That won't happen till car is running.

Reset the oil light, the instructions will be in your owner's manuel

If it flickers at idle, it "might" be engine wear, especially if you are approaching 200,000 miles. It might also be a faulty oil pressure gauge sending unit which is a little more likely.

Fix the dipstick funnel so you do not have oil dripping onto the exhaust causing white smoke. Some people delay fixing a bad oil pressure sender by running a thicker grade of oil. Synthetic oil tends to be thinner , from this you get benefits, in this case it probably exposed an oil sender that is in the process of going bad. If you put a new oil pressure sender in the motor(about$10) you should be able to fix the problem with the oil light.

install a mechanical guage to test the actual pressure, if its fine the sending unit is faulty, if nor remove the oil pan and check the oil pickup for blockage,

see if you have any oil level or pressure sensors. sometimes these can get gunked up. You may need a new oil pump. I suggest you start using a senthetic oil (valvoline,mobil 1 is reccomended) after u get the problem fixed.

change the oil pressure switch or the wrong oil