I agree. It sounds and looks like tires on concrete. Now we would like an apology about Male Chauvinists;)
I think its your starter motor. It sounds like someone turning the key to the start position with the motor running. I had one do this once, it was a rebuild that I had just bought so I just took it back, did not try to find out why.
I have listened for 45 years to women complaining they get treated like an idiot by mechanics or salespeople because they are women. For 45 years, I (a man) have been treated like an idiot. After some study,I finally realized that there is a difference. Men won’t usually admit in public that they got treated like an idiot. They prefer to pretend they really dominated things. It is not true. These people include a lot of jerks, though not all are, and they are just plain ornery to everyone with no regard to race, creed, or gender.
In my life, I have bought a grand total of three new cars. It would have been 8 if the salespeople had not treated me like an idiot. I go; I warn them not to play games, that whatever price they quote is their only shot at me. They play games. I buy a used car from a private party. When they call back four or five days later with a special discount I tell them since their one shot at me was not good enough I bought a used car and if they had given me that price in the first place I’d have bought that car. Tough luck, guys.
After listening to your video it sounds like something in the steering system may be dry, as in not enough grease. Does this happen if the car is not moving and you move the steering wheel? If I were you I would forget about the dealership on this issue and go to an independent mechanic. They are much cheaper and much more “customer friendly”. He can find it for you better than the trained “mentally challenged” people at the dealership.
I can’t tell from the sound on the PC but I might be leaning towards a tire squeak or groan.
Gender has nothing to do with it. There is often a disconnect between the mechanic and customer; usually a service advisor. Peoples’ descriptions of various sounds and symptoms vary just like the story being passed around the Boy Scout campfire. It ends differently than it started.
Another problem is that many people are not mechanically inclined and cannot grasp any explanation no matter what. No, I’m not bad mouthing you here so don’t think that.
I’ve been turning wrenches most of my life and after 35 years with my wife I still go round and round with her on our cars.
I explain in simple details what a car may or may not need and she cannot get it after all of this time. It’s frustrating to discuss a repair/maintenance and then 2 days later listen to her “Comments from Mars” that leave me utterly bewildered and having to correct every other word.
If there’s this kind of disconnect between me and the better half, then consider how it is at the shop. 
As exasperating as such problems can be, my solution has been to charge the minumum shop fee and drive the customers car wherever I needed to go until the problem occurs. No one wishes to pay the hourly rate for a mechanic to determinedly attempt to recreate the problem. Of course, I have kept cars for more than a week before being successfull.
There are chauvanists in my area but the well established shops would never take advantage of a lady but they (we) often struggle to be patient with them (you), and non mechanically inclined men, also. We gear-heads have developed a culture and dialect and sometimes it is difficult to undersstand that everyone doesn’t know the difference between cranking and starting or knowing that the CHECK ENGINE light doesn’t mean to open the hood and see if it’s still there. All car owners should find a shop that treats them fairly and is reasonably patient in explaining problems and solutions… Good luck.
And the winner is…Mr Josh for being the first to come up with the correct answer. I parked the car with the wheels a thin rug. The noise disappeared. I backed off the rug and the noise was back. I also made the noise with the engine off. To me it still sounds like something is getting chewed up, not the quieter sneaker on a wet floor type of noise I associate with tires. It is extremely loud for a tire. But it IS definitely the tires. I am so embarassed.
And I absolutely apologize to anyone I have ever falsly accused of being a chauvinist.
However I will not go back to that dealer. If the independent shops I used to go to haven’t changed ownership I will go there again.
Thank you all for the help, and thanks for not calling me an idiot.
You are correct about service advisors. I have had some extremely good ones. They ask the right questions and don’t insist on twisting my words. (Doctors are really bad about that.). This is so emabrassing for me because I do understand quite a lot. Some other day I will bore the group with the story of our old Datsun and how the dealer kept the car for SIX working days and no one there could figured out why a rear brakes locked up and cause a spectular accident. I was the one who figured out the problem and I had to have a TANTRUM before they checked out the proportioning valve.
I think its time to move on with this. You can’t solve the problem here.
I think its time to retire this subject as its going nowhere. Dear Lady,find a competent mechanic,male,or female and get the issue resolved.
It certainly could be the tires. Have you cleaned the floor recently? It sounds to me as if the tires are turning on sand or dirt. At least my tires sound that way when I need to sweep out my garage. And your garage may amplify sound if the walls are bare. Do check the tires outside, and you might also try the plastic wrap experiment suggested earlier.
Don’t worry about being wrong, if it turns out you are. I have been concerned about a “grinding” noise I hear at 10 MPH in the morning when I first start moving forward. The dealer didn’t know what it was, and I was concerned. It turns out that I hear the ABS pump charging in the morning. None of us knows everything, and it’s OK to ask for help.
If you’re going to go to Indy Mechanics… that EXTENDED WARRRANTY you have is useless.
That being the case, you can CANCEL the warranty, and get a refud for the unused balance!!
Just contact Toyota.