That's right. Orcs are warped from more than goblins, though goblins are their nearest relatives.
Tolkien's Orcs were traditionally (as per the Silmarillion) made from corrupted (by mutilation and necromancy in it's raised from the dead stance - not just talking with them) and deranged elves and various 'spirits', but he later changed his mind and wanted to bring a human strain into it as well - although he somewhat cocked it up because actually the human part of the genetics was supposedly done at much later stages in (officially at least) the Third Age of the Sun with the development of Uruks in Mordor (and possibly Dol Guldur) about 500-1000 years before LotR is mainly based. Saruman definitely bred his orcs from human stock much later than that.
In Tolkien (which is the template for orcs after all) goblins are orcs gone wild and 'diminished' as they're smaller and not as nasty as 'proper' orcs as bred by Sauron and his Dark Lord Morgoth from ancient times. So they're the 'pathetic' Moria orcs who pursue the Fellowship and help capture Merry and Pippin that the Isengard uruks (led by Ugluk) despise and push about, along with the Dol Guldur uruks (led by Grisnakh) who are the original uruks and much nastier than Saruman's lot - in that they're not as disciplined militarily.
Orcs (Yrch) are also called Daugs in Black Speech which means soldiers - that's the difference as orcs are specifically bred for war, whereas goblins are on a par with any other predatory being, but bred back into the mix when times is hard and also as the default retrograde if they escape from their Dark Lord. Nutt's background is with the latter - so ex-soldiers who are gradually going back to the feral state, remembering their origins as soldiers and as the terrible mutated descendants of 'noble' beings (if you can call Discworld Elves noble) who were broken and changed into monsters.
As a domesticated orc people, in their famous way of not noticing anything upsetting, put Nutt down as a goblin because he's so humble and browbeaten, rather than a terrible bloodthirsty full-blown beast of an orc.