Reverts "Reverts "Rename `integration_tests/external_ui` but do not touch anything else..."" (#142268)

Reverts flutter/flutter#142265
Initiated by: matanlurey
This change reverts the following previous change:
Original Description:
Reverts flutter/flutter#142238
Initiated by: eliasyishak
This change reverts the following previous change:
Original Description:
Part of https://github.com/flutter/flutter/issues/142178.