These organisms are categorized as 「使徒」(shito) meaning "apostle" in Japanese sources. Shito is the word used in the Japanese translation of the Christian Bible to refer to people who possess the spiritual gift of Apostle (this differs from the word used to describe all of Jesus' followers, which is 「弟子」 [deshi], the same word used for disciples of Confucius and other renowned historical figures).
Curiously, according to Evangelion Wiki,
The collective grouping has no unique canonical name, although the phrase "Adam's children" was used by Misato in a preliminary draft for Episode 25'
written as 「アダムの子供達」 (Adamu no Kodomotachi), meaning "children of Adam."
According to evageeks.org,
the word "Angel" was retained in the English version of the show at the request of the show's Japanese creators
but they provide no citation for this claim. They note that in an early proposal pitch, the word 「アポストロ」(aposutoro) was suggested.
While apostles are clearly human whereas angels are supernatural, the word "apostle" is derived from the Greek word ἀπόστολος (apóstolos) which means "messenger" (broken apart is "from" + "I send"). The Japanese「使徒」(shito) broken apart is "messenger" + "person" (same kanji as「使う」[tsukau] meaning "use" or "speak"). The special ability of Apostle allows the wielder to go to a new place, preach the Good News there, and plant new church communities with more ease than another disciple who doesn't possess that gifting would be able to do. The English word "angel" is derived from the Greek ἄγγελος (ángelos) meaning "messenger." As you note, the Japanese 「天使」(tenshi) is literally "heaven" + "messenger." So etymologically we see that the Japanese words「使徒」and 「天使」 both show the idea of sent-ness for the purpose of delivering a message.