That sound at the 5sec mark is the sound I hear ... sometimes it cycles a few minutes when starting on cold mornings. I'll try to capture the sound and post a video.
I do not have the other sound that lasts the duration of the video.
My advise: Replace the brake master w/booster assembly. Then Flush & Bleed the system with Toyota brake fluid. Check all brake lines for age related damage/leaks. Replace all 4 rubber bleeder caps, when done bleeding. Tip: Blow out bleeders, before replace rubber caps.
As you video on a cold morning at first key on: You'll very likely see bubbles come up front of reservoir, at same time you hear the squeal
I'll bet your brake fluid reservoir is heavily stained and/or brake fluid is dark and/or darkens soon after flushing fluid?
Why:
A non replaceable (IMHO) seal within the master, is failing. When this seal is cold, brake fluid under pressure passes it. Once it warms by external heat of the day (OAT) or use, it expands and holds pressure. This is why, after using brakes or a day when OAT is high, brakes work as it should. But when cold, the shrunken seal can't hold full pressure. The booster motor will keep running, until the predetermined pressure reached, which signals booster motor to stop.
This seal fails, due to improper brake service. Number one is not regularly flushing. There is also concerns of which brake fluid used. Personally I only use Toyota brake fluid, out of abundance of caution.. Not even Toyota or Lexus Dealership, use their own brake fluid. They're seeing a very high failure rate across all makes!
The brakes will still work, for now! But over time/use, seal will get worst. Booster motor will run longer and longer. The booster motor is designed to run often, but only short duration. Booster motor running longer duration, will cause its premature failure. At some point, booster motor will fail to run. YOU'll LOOSE BRAKES, due to lack of brake pressure! You'll actually have about 2% to 20% braking, at bottom of pedal, with very hard foot pressure on pedal.