The short answer is "yes": an αρσενοκοιτης is most certainly a homosexual. Dr. F. Wilbur Gingerich's abridgement of the longer BDAG lexicon translates the term as "a male homosexual, sodomite, pederast" (pg. 28, first edition). The longer answer is that the term's basic meaning was never in doubt; but certain scholars tried to make distinctions between homosexual "orientation" and certain kinds of homosexual actions in their glosses of the term. Specifically, the question was whether αρσενοκοιτης referred to abusive, or to consensual homosexual "relationships"; i.e., whether Paul was condemning homosexuality in itself, or if he was condemning the sexualization of minors so historically prevalent in Greek culture. This article by Mark Ward is the best summary I've seen on the topic. Essentially, he concludes that the broad term "homosexual" is the best gloss, despite the fact that it does not distinguish between one with same-sex attractions, and someone who acts on those feelings. From the Old Testament teachings on the topic, and the essential continuity of the Testaments, it is right to conclude that our Lord condemns all homosexuality--as indeed He condemns all sexuality outside of covenanted marriage between one man and one woman--as sin, and calls upon those who commit it to repent, and trust in His grace for forgiveness. The Story of Ἀρσενοκοίτης according to BDAG | By Faith We Understand

Hello! I believe I have a solution to your question. Bishop J.B. Lightfoot translated the passage you quote as follows: "When then I had crossed the river, I came into the level country, and knelt down, and began to pray to the Lord and to confess my sins." The reason Lightfoot can translate τιθω as a present indicative verb rather than as a subjunctive, is that there was an alternative form of τιθημι spelled τιθεω in the uncontracted first-person singular present active indicative. This is the form you saw, in its contracted form τιθω. In his book, An Exegetical Grammar of the Greek New Testament, W. D. Chamberlain observes that "there was a gradual process throughout the history of the Greek language of replacing [μι-verbs] with ω-verbs" (87). Given that Hermas was written many decades later than the New Testament, it makes sense that an alternative thematic form of τιθημι should appear in its text, while not being used in the NT. (BDAG mentions the existence of the later form, since it covers both NT and patristic Greek; while some NT-only lexica do not.) A parallel case to τιθημι/τιθεω is that of the verb 'ιστημι, which had the alternative lexical form 'ιστανω in the present active indicative, demonstrating the regularization process which Chamberlain spoke of. I hope this post has been of help!