Try this:
Tools > Board > Boards Manager
Wait for the downloads to finish
Scroll down until you see "esp8266 by ESP8266 Community". Note the installed version shown.
Repeat the process with the other computer.
I believe the one with the Tools > Upload Using menu has an outdated version of the ESP8266 package. Look to the documentation of that package for instructions on using the new OTA system:
https://arduino-esp8266.readthedocs.io/en/latest/