Two screens and three clicks to wake from sleep - can I shorten this?

Hi. My house has two computers with Windows 8.1 that got the upgrade to Windows 10. When waking from sleep in Windows 8.1, one always just woke to its previous state, just the way I like it. The other one woke up to a picture with the clock screen, and had to be clicked past to get back to the state I told it to sleep in.

Now that it is in Windows 10 the first computer does the same thing. The second one with that waste of time clock screen that has to be clicked past now has the same clock, but also to click on which user is going to use the unsleeping windows device.

It's that second computer that's annoying. I never had to click to confirm the user before. Why now?

I don't know what these screen names are called, I keep seeing tutorials like "disable the lock screen" that don't work because it's not locked.

But now it is a three-click workflow.
Click once to wake
Click once to get past the clock
Select the same user that I was before and click to go to the desktop

I know it is possible just to "click once and go" to wake from sleep because I do it every day on computer #1. So what can be done to change computer 2 to make it work the same way?

Thank you.

Thank you Kari for the incredibly detailed cannot-possibly-fail directions that worked perfectly! Everyone in the household is happy now! I even got a chuckle out of the in-forum advertisement about insomnia that was inserted in my thread about a computer "waking from sleep" :) so this was the best possible web experience.