I've now done some experimentation by calling my iPhone from my landline several times, and I may have it figured out.
I was not using the iPhone when the first call came in, and in that circumstance, when I use the slider to accept the call, it takes me to the screen that has the speaker icon (among others).
I think I was using the iPhone on another app when the second call came in, and after I answered the call, I tried to get to the speaker icon by tapping the phone icon. That takes me to the keypad with no "hide" option, and thus no way to get to the speaker icon. If, instead of tapping the phone icon, I tap the green bar at the top of the screen, that takes me directly to the screen that has the speaker icon.
I wish the available documentation was better at explaining this stuff.