Though I am a linguist who has worked on several languages over the years, one of the languages (or language groups) that I have spent the most time studying is Triqui. There are three major Triqui languages (Copala, Itunyoso, and Chicahuaxtla) and though the latter two have some degree of mutual intelligibility, the Copala dialect/language is mostly unintelligible to speakers of the other two dialects/languages.
There are all sorts of interesting things about these languages and about indigenous languages in Mexico, more generally. However, one of the persistent questions I get asked is about the name of the language itself - "is it Trique [ˈtʰɹike] or Triqui [ˈtʰɹiki]?" The answer to this is rather simple - in Spanish used by both Triqui speakers and non-Triqui speakers in Mexico, it's [ˈtɾiki]. So, the closest equivalent in English is [ˈtʰɹiki], with a final [i] sound.
But the follow-up question is usually "Why is it spelled with an "e" then?" To understand this, it's necessary to understand a little bit about dialectal differences in the languages and linguistic practice into the 20th century. To begin, the name of the language ostensibly comes from a spanification (or castellanización) of the Triqui phrase /tʂeh³ (k)kɨh³/, 'father/padre + mountainside/monte', meaning something like 'father of the mountain' in the Chicahuaxtla dialect, though this is a bit debatable. There is another word /tʂːeh³²/ (Itunyoso) or /tʂeh³²/ (Chicahuaxtla and Copala) meaning 'camino' or 'road/path.' So, the name itself may have come from a phrase meaning 'the path of the mountainside.'
One thing to notice is that the Chicahuaxtla dialect retains the central vowel /ɨ/ where it has merged with /i/ in the Itunyoso dialect and, in some contexts, with /u/ in the Copala dialect. So, the word for 'mountainside/monte' retains this vowel in Chicahuaxtla where the word is /kːih³/ in Itunyoso Triqui and /kih³/ in Copala Triqui. This vowel also exists in many Mixtec languages (Triqui is Mixtecan) and is reconstructed for Proto-Mixtec (Josserand, 1983).
The first Triqui language to be described was the Chicahuaxtla dialect (Belmar 1897) and he wrote the name of the language as Trique. Now, Belmar was not particularly adept at transcribing many of the nuanced phonetic details of many languages. His tonal transcription is non-existent and he misses many important suprasegmental contrasts. However, he chose "e" here because he heard a difference between /i/ and /ɨ/ and it was customary at the time to transcribe this latter vowel with "e." This practice goes back to very early Mixtecan/Otomanguean philology - the Dominican friar Antonio de los Reyes (1593) used "e" to transcribe this vowel in Teposcolula Mixtec. So, the six historical Mixtec vowels are, at least in old historical sources, transcribed as /i/ "i", /e/ "ai", /a/ "a", /o/ "o", /u/ "u", /ɨ/ "e." The IPA certainly did not exist during Belmar's time and this practice is simply an extension of a Mexican philological tradition.
Incidentally, the use of 'e' for transcribing mid back unrounded vowels is not limited to languages in Mexico. The romanization of Chinese, called pinyin, uses "e" for the vowel /ɤ/, found in many Chinese languages. This practice in fact seems to go back to earlier romanizations of Chinese and, in fact, the earliest grammar of Chinese was Arte de la lengua Mandarina, written by another Dominican friar, Francisco Varo. Though, as far as I can tell, he did not use "e" in his romanization of Chinese - that came later.
The earliest work on Triqui written in English is Longacre (1952) and he must have simply taken the practice of writing the language with an "e" from Belmar and other Spanish sources. Nowadays, it is written with an "i" in Spanish. Though, due to older sources using an "e", such as all work by Hollenbach on the Copala dialect, from 1973 to 1992, the spelling with an "e" has stuck around.
Belmar, F. (1897). Lenguas del Estado de Oaxaca: Ensayo sobre lengua Trique. Imprenta de Lorenzo San-Germán.
Hollenbach, B. E. (1973). La aculturación lingüística entre los triques de Copala, Oaxaca. América Indígena, 33:65–95.
Hollenbach, B. E. (1992). A syntactic sketch of Copala Trique. In Bradley, C. H. and Hollenbach, B. E., editors, Studies in the syntax of Mixtecan Languages, volume 4. Dallas: Summer Institute of Linguistics and University of Texas at Arlington.
Josserand, J. K. (1983). Mixtec Dialect History. PhD thesis, Tulane University.
de Los Reyes, F. A. (1593). Arte en Lengua Mixteca. Casa de Pedro Balli, Mexico, Comte H. de Charencey edition.
Longacre, R. E. (1952). Five phonemic pitch levels in Trique. Acta Linguistica, 7:62–81.