Though the wrong notion that advaitins are saivaites has gained currency, the opinion that 'advaitA alone is the complete (pUrna) vEda matham' had also been present, particularly after the
establishment of various siddhAnthAs by the later day AchAryAs, which differed from advaita. The continuation of the common title - smArthAs, only for the advaitins [*till today*] is a proof of this.
The root meaning of the word 'smArthA' is neither 'the one who had adopted advaita siddhAnthA' nor 'the one who follows ShankarA'. But, it refers to the one who follows the completely 'vEdic smrithis' (dharma shAstrAs). Similarly, the surnames such as 'sharmA, shastry' etc have continued only for smArthAs.
Even from the shAstric works, which were in vogue long before theadvent of AchAryAl, it becomes known that 'sharmA' is referring to the bhramin caste. It is a tradition from vEdic times to refer to a bhramin as 'sharmA' and a kshatriyA as 'varmA'. If we say that there was a painter by name Ravi VarmA, then it implies that he was a kshatriyA.
Even now, when a bhramin boy prostrates, after his upanayanam, by telling his vEdic sAkhA, sUthrA and gOthrA, we ask, "what is [*your*] sharman?". Even he tells his name as 'such and such sharman'. This is applicable even for iyengArs and mAdhwAs.
But, the one who always uses this 'sharma' as a surname, indicative of his caste, will invariably be an advaitin. This shows that he is the only complete (pUrna) follower of vEdic matham. 'ShAstry' means the 'one who follows shAstrA-s'. [*The word*] ShastrA refers only to vEda-dharma shAstrA-s. Even this is used as a surname only by the advaitic smArthA.
As an exception, either a mAdhwA or a Vaishnavaite may rarely have 'sharmA' as his surname. But, in general, by the mere mention of names such as 'D.S. Sarma' and 'Right Honourable Shastry', we infer that these [*persons*] are only smArthAs.