Yes, that explanation is correct—for most of them.
Examples of scholarly referencess on the subject include 1. A Dictionary of English Surnames and 2. The Oxford Dictionary of Family Names in Britain and Ireland.
As the OP suggests, most last names that end in -smith are 'occupational last names', where the smith typically means, roughly, 'maker of'. But there are some exceptions: Goodesmith/Goodsmith, Hudsmith, and Athersmith. Silversmith is a different kind of exception; see below.
In Goodesmith/Goodsmith and Hudsmith, the root is not really smith, but rather
ME maugh, mough which may be from ON mágr ‘brother-, father- or son-in-law’, OE māga ‘relative, son’, OE māge ‘female relative’, or OE magu ‘child, son, servant’.
See e.g. the entry for Hudsmith here.
In Athersmith, the -smith comes from one of two sources. First, it may denote 'someone who lived, and presumably worked, at a smithy (Old English smiþþe)'. The second possibility is that it comes from OE smēðe ('free from roughness, not harsh, polished; soft; suave; agreeable'; see here), in which case it is
a locative name possibly also from Middle English at ther smethe ‘dweller at the smooth, level place’'
In either case, unlike most other -smiths, Athersmith does not come from 'a maker of something', but it rather has to do with where the person lived.
At the end of the day, this is an 'occupational' surname just like Goldsmith and other -smith occupational surnames I list below. However, the historical origin of Silversmith is very different from that of these other surnames.
Shockingly enough, unlike the surnames listed below, Silversmith seems to not have been used as a surname in pre-modern British isles. Instead, it seems to come from America, as a 'part-translation of German and Ashkenazic Jewish Silberstein'. I base this claim on this entry from Dictionary of American Family Names, and on the fact that neither the two scholarly references above nor the 1860 Patronymica Britannica, a dictionary of the family names of the United Kingdom have an entry for Silversmith (see here). This despite the fact that all three of these references do have Goldsmith, which is indeed an occupational surname with a long history of presence in England, just as you'd expect.
A list of -smith surnames in A Dictionary of English Surnames
Here are all -smiths that have entries in A Dictionary of English Surnames. The abbreviation 'OE' means 'Old English'; 'ME', 'Middle English'; and 'ON', 'Old Norse'.
Arrowsmith, Arsmith, Harrismith,
: Roger le Aruesmuth 1278 AssSt; William le Arwesmyth 1324 FFEss; Richard Arsmith Eliz Bardsley. OE arwe ‘arrow’ and smith. A smith who makes arrows, especially iron
arrow-heads (1278 MED). Harrismith and Harrowsmith are rare but exist side by side with Arrowsmith in Yorks and Lancs.
: John atter Smythe (1330 PN D 386) lived at the smithy (OE smiþþe) and was
presumably a blacksmith. The surname might also derive from ME at ther smethe ‘dweller at the smooth, level place’ (OE smēðe). cf. William del Smethe 1327 SRSf.
: Thomas le Brounesmyth 1296 Wak (Y); William Brounsmyth 1327 SRSo. OE brūn and smið ‘brown smith’, ‘a worker in copper and brass’.
: Richard Copersmid 1212–23 Bart (Lo); Robert copersmith eHy 3 ib.; John le copersmyth 1305 LoCt. ‘Maker of copper utensils’ (1327 MED). cf. Hugo Coperman 1202 P (We), Stephen le Coperbeter 1286 LLB A.
: Roger Goldsmiz 1250 MESO (Nf); Thomas Goldsmith 1255 Ass (Ess); John le
Goldesmethe 1309 LLB D. OE goldsmið ‘goldsmith’.
: John Godesmaghe 1351 FrY; Margaret Goodsmyth 1525 SRSx. ‘Relative of Gode’, cf. HUDSMITH.
: Thomas Grenesmyth 1523 RochW. ‘Coppersmith.’
: v. ARROWSMITH
: v. ARROWSMITH
: Thomas Huddemogh 1332 SRLa; William Hudmagh 1379 PTY; Thomas Huddesmawth 1464 FeuDu; Cuthbert Hodgemaght, Hodgemaughthe 1545–6 NorwW (Nf); Ralph Hudsmyth 1582 PrGR. ‘Hudd’s brother-in-law.’ cf. Richard Gepmouthe and v. WATMOUGH.
: Walter, Roger le Loksmyth’ 1255 Ass (Ess), 1293 Pinchbeck (Sf); John Loc Smyth 1279 AssNb; Robert Locsmyth 1279 RH (Hu). OE loc ‘lock’ and smið, ‘locksmith’. cf. William Locwricht 13th AD iv (Lo).
: v. NAYSMITH
: v. NAYSMITH
Naysmith, Naismith, Naesmith, Nasmyth
: Roger Knifsmith 1246–89 Bart (Lo); Adam Knyfsmith 1285 AssLa; Saman le Knyfsmyth 1310 LLB D; William Knysmyt 1326 AssSt; Robert Knysmithe 1594 Bardsley; John Naysmith n.d. ib. OE cnīf ‘knife’ and smið ‘smith’, a cutler.
: John Platesmyth 1379 PTY. ‘A maker of plate armour’, ME plate, OE smiþ.
: v. SHEARSMITH
: Walter Scheresmythe 1325 ParlWrits (Gl); Geoffrey Sheresmyth 1391 FrY. ‘A maker of shears, scissors’, OE scēarra and smið.
: William Le Shosmith, Sosmyth 1288 MESO, 1296 SRSx. OE scōh, smið ‘shoeingsmith’, maker of horseshoes.
: v. SUCKSMITH
Smith, Smithe, Smyth, Smythe, Smye
: (i) Ecceard Smið c975 OEByn (Du); Ælfword þe Smith c1100 ib. (So); William le Smyth 1275 AssSo; Julian’ le Smithes 1279 RH (O); John Smye 1524 SRSf. OE smið ‘smith, blacksmith, farrier’. Early examples are common in the Latin form FABER. (ii) William atte Smithe, Thomas de la Smythe 1313 AD i (Sx); Robert atte Smyth 1332 SRSx. ‘Worker at the smithy’, OE smiþþe.
: ‘A maker of plough-shares’, OFr soc, OE smiþ.
: v. WILDSMITH
: v. WILDSMITH
: Richard, William le Wytesmith 1260 Cl (O), 1279 RH (C); John le Wytesmyt, le Whitesmyth 1313, 1332 SRWa. OE hwīt and smið ‘white-smith’, a tin-smith (1302 NED).
Wildsmith, Wilesmith, Weldsmith, Whilesmith
: Euota Welsmyth’ 1319 SR (Ess); Ivo le Welsmyth 1327 SR (Ess); Anne Wilesmith 1787 Bardsley. This surname has been variously explained as a corruption of weld-smith, a forger in iron, weald-smith, and the smith in the wild, none of which is satisfactory, nor is Bardsley’s association of the name with wool. Welsmyth is from OE *hwēol-smið ‘wheelsmith’, a maker of wheels, especially the iron parts. cf. WHEELWRIGHT. The vowel was raised to [i] (Wheelsmith), then shortened to Willsmith, which, with an intrusive d, became Wildsmith. ME W(h)īlsmith developed normally to W(h)ilesmith. cf. Whygler for Wheeler, Whilwright for Wheelwright, and the development of weald to wild.
There is no entry for Wordsmith in any of the four dictionaries mentioned above. However, I only have access to the American one through google books, and so it is possible that it has an entry for it after all, but that we just can't see it.
On the related topic of surnames that begin with Smith- (other than Smith, which I already included above):
: Robert de Smetham 1275 RH (D); Thomas Smitham 1642 PrD. From Smytham in Little Torrington (D).
: John Smythiere 1379 AssWa. ‘Smith, hammerman.’ cf. ‘The Jorneymen…of all oþer Craftes…except hakmen and smythers wurche in hur own houses and nott in hur masters housz’ (1435 NED).
: Robert Smythyman 1309 Cl (Y). ‘Smithy man’, worker at the smithy.
Smithies, Smithyes, Smythyes
: John del Smythy 1332 SRLa, 1385 ShefA; John Smythes, Smythies 1568,1586 FrY; Thomas Smiddies 1629 ib. ‘Worker at the smithy’, ME smythy.
Smithson, Smythson, Smisson
: Reginald le Smythessone 1296 SRSx; Henry le Smithson 1324 Wak (Y); Peter Smitson 1327 SRY. ‘Son of the smith.’
Smithwhite, Smorthwaite, Smorflt, Smurtbwaite, Smurfit, Smallthwaite
: Henry de Smetwayt 1285 AssLa; Thomas de Smythuait 1327 SRY; Matilda Smarthwate 1489 GildY; Robert Smerwhaitt 1518 ib.; Thomas Smirthwaite 1577 FrY; Thomas Smorthwait 1627 ib.; Matthew Smurwhaite 1657 ib.; Edward Smurfett son of Matthew Smurfett 1713 ib. In Cumberland there is a Smaithwaite in Keswick and another in Lamplugh, Smathwaitis c1280, Smerthwayte 1530, Smethwayte 1552 (PN Cu 314, 407), whilst in Guiseley (Y) we have Smerthwayt 1323 Calv, ‘small clearing’ (ON smár). In Cumberland, too, we find two Smallthwaites, neither recorded before 1611, which may be identical in origin, with late substitution of English small for Scandinavian smár. The first two examples above may contain OE smēðe ‘smooth’.
: Adam de Smithewyk 1327 SRSx; Roger de Smythewyk 1332 SRSx. From Smethwick (Ches), or from a lost Smithwick in Southover (Sussex), last recorded c1608 (PN Sx 323).
: v. SMITHERMAN
On the general topic of how these things are determined, read the introduction (and the prefaces) of the Dictionary. Here is the very beginning of the introduction:
The purpose of a Dictionary of Surnames is to explain the meaning of names, not to treat of genealogy and family history. The fact that Robert le Turnur lived in Staffordshire in 1199 and that there was a William de Kouintre in London in 1230 does not mean that they were the ancestors of all or any of the modern Turners or Coventrys. To establish this, a fully documented pedigree would be required and very few families can carry back their history so far. Throughout the Middle Ages surnames were constantly changing. William Tyndale was known as Huchyns when living in Gloucestershire. Oliver Cromwell was a Williams and David Livingstone was a McLeay. Even today families change their names. Blackden has become Blacktin, Hogg has been changed to Hodd and Livemore has superseded Livermore—all within living memory.
The modern form of many of our surnames is comparatively recent, often preserving a phonetic spelling found in a seventeenth- or eighteenth-century parish register. When some of the Sussex Bourers migrated to Kent in the seventeenth century they adopted the form Bowra. Pharaoh is a reconstructed spelling of Faro, originally Farrer, found also as Farrey, Farrah and Farrow in the seventeenth century. The Suffolk Deadman is a corruption of Debenham and Tudman of Tuddenham. Each surname has its pedigree which must be traced before the meaning can be discovered, and even then the true origin cannot be decided unless the family pedigree can be carried back far enough to fix definitely the original medieval form. A modern White may owe his name to an ancestor bearing the Anglo-Saxon name of Hwīta, or to one nicknamed ‘the fair’, or to an original home in the bend of a river. The original Howard may have been a ewe-herd or a hayward, or he may have borne either the French name Huard or the Old German name Howard. The modern forms often conceal rather than reveal information.
The English language lacks terms corresponding to the French sobriquet and nom de famille. Today, surname means an inherited family name; originally it meant simply an additional name and it is used in this sense in this book. Only very occasionally can early medieval surnames be proved to be hereditary, and any attempt to distinguish them would end in inaccuracy and confusion.