Probably not the pump. Sounds like multiple problems, and scale related. When was the last time the machine was descaled (not backflushed)? The water leaking out is definitely something that needs further investigation: water + (electronics and electricity) = bad thing.

Could be a scaled water level sensor, scale in the pressurestat, a problem with the water solenoid or related electronics, a stuck vacuum breaker, faulty heat exchanger, stuck pressure relief valve, or other such stuff... It sounds like you are not mechanically inclined (forgive the assumption). If that is true, sounds like time for a professional service.

Let's back up for a moment...

Nearly all HX espresso machines have one pressure gauge for the steam boiler. They range from 0 to around 2 bar and normally operate around 1.0 to 1.2 bar when fully heated. Many HX espresso machines have a second pressure gauge that indicates pump pressure. They range from 0 to around 12 bar and normally operate around 9.0 bar when the pump is running during a shot.

From what I understand of your description, the steam boiler is overheating ("pressure gauge started to push into the red zone"). That indicates the pressurestat, which regulates the steam boiler pressure, is stuck in the closed position or failing to read the correct pressure. A safety valve will release steam to prevent the steam boiler from going "boom"; these valves typically open around 1.6 to 2.0 bar. That would explain the large water puddle on the counter.

There's lots of threads explaining how to diagnose, and if necessary, replace the pressurestat ("sticky pressurestat"). If you're moderately handy, comfortable with the idea of DIY repairs, and agree to the site's terms of service on this point, the information is readily available. That said, I agree with Randy that a professional repair technician may be the best course of action for the same reasons he cited (forgive the assumption).

IMPORTANT: If the steam boiler safety release has opened or the machine is dripping water, turn off and unplug the machine immediately! The interior is potentially flooded with water condensation, which represents an electrical shock hazard. Water dripping in the interior could also short out some very expensive electrical components, if it hasn't already based on your comment ("After turning off the machine and cleaning everything I could think of, I turned it on again and nothing"). If you're lucky, you've only tripped the electrical outlet GFI.