I think the other two answers miss the question. The way I interpret the question is "E.g New Mexico is on Mountain time, and observes DST, Arizona is in the same time zone, but does not observe DST, what term would be used to differentiate the two? And what terms differentiate the period using DST from the period not using DST".
There is no concise term used to describe this other than to say one place observes DST and the other does not.
Rather, the name of the time observed actually changes, e..g from Wikipedia's "Terminology" section
The name of local time typically changes when DST is observed. American English replaces standard with daylight: for example, Pacific Standard Time (PST) becomes Pacific Daylight Time (PDT). In the United Kingdom, the standard term for UK time when advanced by one hour is British Summer Time (BST), and British English typically inserts summer into other time zone names, e.g. Central European Time (CET) becomes Central European Summer Time (CEST).
But the original Standard Time Act of 1918 specifically indicates that it is the standard time which changes (emphasis mine):
Sec. 3. That at two o'clock antemeridian of the last Sunday. in March of each year the standard time of each. zone shall be advanced one hour...
And goes on to define the names for each time zone, introducing no terminology to differentiate DST from standard time:
Sec. 4. That the standard time of the first zone shall be known and designated as United States Standard Eastern Time; that of the second zone shall be known and designated as United States Standard Central Time; that of the third zone shall be known and designated as United States Standard Mountain Time; that of the fourth zone shall be known and designated as United States Standard Pacific Time; and that of the fifth zone shall be known and designted as United States Standard Alaska Time.
So, the best (legally correct) term to use to identify a time system of a place that observes DST is just "Standard Time". But that would obviously be confusing it today's world.
The term used to describe the period of time when DST is observed is often referred to as "The DST Period" (National Conference of State Legislatures). And the period not on DST is just "Standard Time".
Probably the most widely used reference for time zones is https://en.wikipedia.org/wiki/Tz_database. From its List of Time Zones, you'll see that each time zone is given a name, such as "America/New_York" to describe the overall time zone, and an abbreviation which designates DST or Standard Time, and uses the terms STD for standard time, and DST for DST. And each has an entry for its UTC offset.
So, the two best fill-ins for the blank in your question
"What _____ are we using today? Daylight Savings Time or Standard Time?" would be either "Period", or "Time Zone Abbreviation". I think we'll all agree that neither would go over well in a typical conversation, and it'd be better to just ask "Are we on Standard Time or DST?". And this is essentially how nearly all systems that observe DST work. They allow the user to specify a Time Zone, then have a flag to indicate if the user wishes to observe DST.