A separated "How To" thread with instructions that we can reference when someone needs help with downloading styles.

1. How to download styles using a USB stick.
When you download a zip file you need to unzip it first. Can either right click on it and select "extract all", or double click to get into the archive, then drag the file out with a mouse or use ctrl+c/ctrl+v and copy it to a USB stick.
After a USB stick is inserted into a keyboard you need to click on a style selector (as if you are selecting a different factory style), then click on User menu at the top (next to Preset), then click on USB, and should see a style there.

2. How to load a style though an USB cable:

Install "USB-MIDI Driver" from Yamaha (currently v.3.1.4)
Use USB-B cable (square shaped) and insert it to "To Host" port on your keyboard
Make sure the cable is connected before you turn on the keyboard (if already turned on, need to turn off and on)
Open a browser (Chrome/Firefox work the best, IE is not stable, Safari doesn't work) and go to the style download page on this site (if you had the browser already opened before you connected the keyboard it's better to restart the browser)
The first time you visit the style download page a popup should appear asking to give the site permissions to access your midi devices, need to allow it. If you missed this popup the first time, look for a keyboard looking icon (black and white piano keys) in your browser's URL bar to open it again.
Your keyboard should be selectable now under midi in/out in the "transfer" section. You need to select your keyboard under both In and Out (it might have a generic name like digital workstation 1). Usually you would have 1 input and 2 output ports available in a keyboard, need to select the 1st one as input and the 1st one as output.
After that if you did everything in correct order the file should transfer and you should see a green confirmation popup listing under what filename it was saved.
There are two options for transferring styles to your keyboard: Preview and Transfer. Preview loads a style into keyboard's memory without storing it on your keyboard drive. It is useful for quickly trying styles without worrying about deleting them afterwards. And Transfer saves a style into USER section of you keyboard.
To find a transferred style inside your keyboard need to go into the style selector (click on a style name), then switch to User section at the top (next to Preset), and you should see it there.

To test if your keyboard is connected correctly and that the browser can communicate with it can try these midi event monitors:
https://hardwaretester.com/midi
https://www.midimonitor.com/

Currently this method works for PSR-SX series and later.
It is also confirmed to work on Android through Chrome via a USB cable. Whether it works on Android wirelessly through Bluetooth is unknown.
It doesn't work on IOS. It works on Mac through non-Safari browser.