MGM CTLR REPORTED ACFT FLEW WRONG APCH PROC TO ALX WHEN IN FACT CLRNC ISSUED WAS NOT COMPLETE AND/OR ACCURATE.

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IFR ACFT WAS HANDED OFF TO ME INBOUND TO ALX WHICH IS A NON-TOWERED ARPT. I INSTRUCTED THE ACFT TO EXPECT THE GPS RWY 36 APCH. THE PLT ASKED FOR VECTORS TO FINAL. I TOLD HIM TO EXPECT THE VECTORS BUT CONSIDERED IT ANNOYING BECAUSE IT WOULD HAVE BEEN MUCH EASIER FOR HIM TO GO DIRECT TO TATME (STRAIGHT-IN IAF). AT ABOUT 15 NM SSE OF THE ARPT; I CLRED HIM '14 MI FROM BILLP; MAINTAIN 3000 FT UNTIL ESTABLISHED ON THE FINAL APCH COURSE; CLRED GPS RWY 36 APCH.' A MIN OR TWO LATER; JUST E OF TATME; HE TOLD ME THAT HE HAD BROKEN OUT AT 2200 FT. I ASKED HIM TO VERIFY THAT HE WAS ESTABLISHED ON THE FINAL APCH COURSE (SINCE HE APPEARED TO BE E OF TATME) AND HE REPLIED THAT HE WAS. I TOLD HIM THAT THE CHART INDICATED AN ALT RESTR OF 2500 FT UNTIL THE FAF. HE REPLIED THAT HIS CHART INDICATED 2200 FT. I NOTICED THAT THE NDB OR GPS-A APCH HAD A 2200 FT ALT; SO I ASKED HIM TO VERIFY THAT HE WAS FLYING THE 'GPS RWY 36 APCH.' HE SAID; 'AFFIRMATIVE; THE GPS-A APCH.' I ADVISED HIM THAT THE GPS 36 AND THE GPS-A WERE DIFFERENT APCHS AND HE WAS NOT FLYING THE ONE THAT HE HAD BEEN CLRED FOR. HE APOLOGIZED AND SAID HE WAS VFR AND COULD CANCEL HIS IFR FLT PLAN; WHICH HE DID. IN THIS PARTICULAR CASE; I DON'T THINK SAFETY WAS COMPROMISED BECAUSE HE DID FLY A VALID; SERVICEABLE APCH AND NO OTHER ACFT WERE A FACTOR. IT COULD HAVE BEEN A PROB IF OTHER ACFT WERE NEARBY OR IF I HAD CHOSEN AN APCH BECAUSE THE OTHER WAS NOTAMED AS NOT AUTH. UPON REVIEWING THE ISSUE TO FIGURE OUT HOW I COULD DO BETTER; I CAME TO THE REALIZATION THAT AN APCH NAMED RNAV (GPS) RWY 36 SHOULD PROBABLY BE REFERRED TO AS THE RNAV RWY 36 APCH; EVEN IF THE PLT IS USING GPS TO FLY IT. THIS IS A SOMEWHAT SUBTLE DIFFERENCE FROM ONE LABELED NDB 'OR' GPS. I WILL CHANGE MY TECHNIQUE NOW TO CALL THESE RNAV (GPS) APCHS 'RNAV' INSTEAD OF GPS.