As the Jewish Encyclopedia states,
to the middle of the nineteenth century no attempt was made to elaborate a scientific system of transcription of Hebrew in foreign characters, and every one followed his own caprice. In 1854 Bargès published the Book of Ruth with a French transliteration of the text. In his system, which was followed by nearly all the French Orientalists, the letters בגדכפת, according as they have or have not dagesh, are represented by their equivalent French letters with or without "h."
Even an earlier source from 1821, Moses Stuart's grammar uses the same principle:
When any of the letters בגדכפת are written without Daghesh lene, they are said to be aspirated; e.g. ת = th, or has the aspirate h united with it. When they are written with a Daghesh lene in them, they are said to be unaspirated; e.g. תּ = t, or is written without the aspirate h.
However, when JE cites Bargès a bit later, they fail to transliterate the citation according to his principles (see מִבֵּית). This may be due to the fact that the JE's simplyfied system didn't differentiate between ת and תּ, which they based on recommendations of the Geneva Congress of Orientalists.* There were many authors and journals, who did this distinction, while there were some others that didn't (see the tables in Werner Weinberg's paper). To illustrate the randomness, the American Library Association used only t in 1908, then both t and th in 1941, and reverted back to only t in 1949, while still having inconsistent examples (Japheth vs. Sheshet).
Although one is clear, all these methods follow the Sefardic pronunciation (no s for ת). Almost none of the English books had the Ashkenazi system (aulom, aylom, oylom etc. for עוֹלָם), but were using the simplified one proposed by the JE either only with t or with t and th. Two extremely influential 20th century books in the English-speaking Jewish world, The Pentateuch and Haftorahs edited by Hertz and the Soncino Talmud edited by Epstein had the t/th system, just as in your examples.
Even though now it's almost impossible to reconstruct the intent of individual authors, Bargès made clear that without dagesh he proposed a soft and aspirated sound, while Weinberg (p. 4.) is more explicit by giving a good example:
When a congregation called itself "Adath Yeshurun," [...] the rendition of ת by th could be either a German transliteration (in old orthography) to be pronounced /t/ or an English transliteration-transcription, rendering the pronunciation of undageshed ת as that of unvoiced English th. The original intention was then to realize the pronunciation /adát (or adáθ) yəšurún/. But the congregational member will usually pronounce it the way the spelling suggests, namely /ædəθ ǧéšərən/.
Indeed, the latter th is closer to the original Tiberian pronunciation (which is preserved by the Yemenites) and was strongly advocated by Sefath Emeth.
* It was in 1894, but I couldn't find any reference on this issue in the report.
