Thank you for your service. Mine ended before the GWOT.
https://apps.dtic.mil/sti/citations/ADA466395
Defense Technical Information Center
Title: Gulf War II: Air and Space Power Led the Way
https://www.airandspaceforces.com/article/0503verb/
Ar and Space Forces Magazine
Title: Verbatim Special: Gulf War II
http://www.ipcs.org/comm_select.php?articleNo=1017
Institute for Peace and Conflict Studies
Title: Gulf War II: Some Reflections
I have no idea how old you are, but I remember vividly that when the Global War on Terror started after 9/11, the invasion of Iraq was commonly referred to as "Gulf War II."
I grant you that after 20 years of occupation, the moniker has fallen into disuse, and I also agree that it wasn't an "official" title of any campaign, but as you can see from both civilian think tanks and military sources I have cited above, Gulf War II was in use by more than just pundits.
I was in the Service the day Saigon fell and I retired in 2008, I served as a contractor in country and as an instructor for foreign military services. All in all I began with the DOD in 1974 and ended my time with the DOD in 2016. Because a pundit decides to use catchphrase does not mean that the DOD or other three letter fed agencies considered the Gulf War as being linked to GWOT, OIF, OEF or Noble Eagle. I have heard the phrase used on CNN and FOX that does not mean it has any validity. We won the Persian Gulf War we successfully completed our mission. Like Nam we won the battles but lost OIF and OEF.