Sounds like something on the front of the motor since it is speed dependent. Could be belt tensioner, power steering pump, alternator, air conditioning pump or water pump. But since you say that it is having problems starting, I would start with the starter itself. Take it off and have it tested.
x2 what USMC said. Just guessing over the interweb I would say PS pump or Alternator.
Maybe try taking a video while stopped and with the hood up, walking from one side of the engine bay to the other to try and pinpoint which area the noise is coming from.
agreed: check the tensioner pulley for wiggles & dont forget that as the temp drops , belts begin to loosen-up ever so much which can exacerbate problems that weren't noticeable when weather was warmer ... belt could even be slipping a bit ? it's a totally different noise after you stop/start it and the 'after start' sounds like a loose idler pulley where the pre-restart sounds like belt slippage .
just my ￠.02
<edit: alternators going bad this time of year (cooler) are very common if it isn't the starter as mentioned >
If it sounds like trans area, check where the exhaust crosses over and make sure it's not laying on the skid bar underneath it. I keep bending that bar up in to my exhaust, but never get a noise like that. Let us know what you find.
An old Mechanic's trick to pinpoint a sound from an engine or hear it better is plug on ear really well and touch the engine (while running) with a long screwdriver. Then put the handle to your unplugged ear. Just pay careful attention to belts, pulleys and other moving objects when you lean into the engine.
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