In the U.S.A. the term Veteran legally does not mean that a member of the military has seen combat, and it never has. The term Veteran has always included military personnel who served in support roles. The term Veteran is codified legally in U.S.A. Federal Code 38 §3.1(d) Veteran means a person who served in the active military, naval, or air service and who was discharged or released under conditions other than dishonorable.

This definition did recently slightly change to it's current state though. In 2016 members of the National Guard were considered veterans only if they served 180 days or more in active federal service outside of training. After the law changed in 2016, if you had served in the National Guard for a period of over 20 years, you could attain Veteran status if you served in active (Federal) duty for any period of time, not just 180 days. This may be what you are referring to in your question about military members who were never mobilized. However they must have been mobilized/activated for federal service for some amount of time. In 20 years of service these guardsmen may also have been mobilized repeatedly by their state for non-combat duties for things like non-federal disaster response, so Veteran as defined in U.S. federal code never has and currently does not cover Guardsmen who were never ever mobilized.

Federal code 38 does have a classification for Wartime Veteran which has additional pension benefits. In Title 38 Code of Federal Regulations for the first period where Wartime Veterans are recognized starts is in 1817:

§3.2 Periods of war (a) Indian wars. January 1, 1817, through December 31, 1898, inclusive. Service must have been rendered with the United States military forces against Indian tribes or nations.

Congress passed a separate service-pension statute for pensions to be paid Revolutionary War veterans in 1818. Being designated as a Wartime Veteran or Surviving Spouse does not necessarily mean that a Wartime Veteran saw combat though. Different period designations have different guidelines, for example:

(h) Mexican Border War: May 9, 1916 through April 5, 1917. The veteran must have served at least one day in Mexico, on the border thereof or in the waters adjacent thereto

Though term "Combat Veteran" is not part of this code, the code does recognize additional pay and benefits for duty in combat zones, or additional benefits for veterans who have wounds or disabilities as result of combat injuries. Public Law 110-181 passed in 2008 additionally ordered long term studies of veterans who experienced combat in order to identify if combat is correlated to increased long-term disabilities, injuries or other issues that are not evident immediately. Also the law grants enhanced V.A. medical benefits for injuries that result from combat beyond what is specified in Federal Code 38. The Public Law 110-181 section is titled TITLE XVI - WOUNDED WARRIOR MATTERS, and does not use the term Combat Veteran. The V.A. though uses the term Combat Veteran as the V.A. in drafting rules and guidelines to impliment the public law had to have a term which was concise and easy to understand. The VA recognizes Combat Veterans as:

Veterans, including activated Reservists and members of the National Guard, are eligible if they served on active duty in a theater of combat operations after November 11, 1998, and have been discharged under other than dishonorable conditions.

The classification of Combat Veteran requiring service in a theater of combat operations does not require participation in actual combat and includes military personnel in support roles.