I followed the instructions, but when I try to run the program, the underlined lines are not recognized and the program fails.
My platformio.ini is
platform = espressif8266
board = esp12e
framework = arduino
monitor_speed = 115200
lib_deps = ESP Async WebServer
Should I install these packages in some way, since I find a bit odd that with only_deps line the packages are installed?
Try to install the libraries manually.
Delete the libraries you already have from the plaformio.ini file.
Then, in VS Code click on the Libraries tab (in PIO Home), search for the libraries you want to include, and then add them to your project.
Try to compile the code again and check if the issue is solved.