Here the original manuscript reads “that same servant”, but Oliver Cowdery, when copying to the printer’s manuscript, accidentally changed the demonstrative that to the definite article the. Further evidence that the original manuscript’s that is correct is found a few words earlier in that manuscript. After the initial words “and it came to pass”, Oliver initially wrote that same se (where se was the beginning of the word servant). Oliver caught his error immediately and deleted same se using erasure and crossout; he then continued inline with the correct text (“Amalickiah took that same servant that slew the king”). In other words, Oliver anticipated the phrase “that same servant”.
Usage in the original text is fairly equally divided between “that same ” and “the same ”, with 32 and 27 occurrences respectively. So either determiner works. In each case, we follow the earliest extant reading; here in Alma 47:34, 𝓞 reads “that same servant”. There is additional evidence that Oliver Cowdery tended to replace “that same ” with “the same ”:
Summary: Restore in Alma 47:34 the demonstrative that in “that same servant”, the reading of the original manuscript.