I guess there are various forms of Dhamma, for example:
- Abhidhamma (and other books like the Visuddhimagga)
- Mahayana teachings
- Contemporary books and Dhamma talks
It was that I had read only contemporary books, i.e. modern authors writing about Buddhism; or, paraphrasing (retelling) doctrine such as the story of the Buddha's life.
Thankfully I found and have been able to read some more-or-less accurate translations of the suttas (see here and here).
After that (having read some translations of various suttas, from various translators on sites such as accesstoinsight, suttacentral, and dharmafarer), it seems to me that I may recognise what sounds like true dhamma. The literature is lengthy (i.e. there are many suttas) but maybe it's fairly coherent, with maybe a dozen or so major topics, and maybe some underlying/unifying theme[s].
I also like to read essays about suttas, or which quote suttas.
One way to recognise adhamma is that it seems to contradict what you'd recognize as dhamma.
One example might be, if it makes an assertion like,
This is a war between righteousness and unrighteousness (dharma and adharma) and it is your duty to slay anyone who supports the cause of unrighteousness, or sin.
... then I'd think that's maybe not Dhamma because it seems to contradict the first precept.
Another way, in my own experience, is based on the literary style. For example here is a list of about two hundred Fake Buddha Quotes -- the author commented on each one, so maybe that's a good introduction/explanation as to how to recognise them.
Maybe don't read them all, but scan or skim the list, and read any that seem interesting or borderline: for example if there are any where you ask yourself, "What's fake about that?"
This answer is incomplete in that I won't try to explain how to recognise dhamma (or adhamma) other than the suttas -- so it doesn't do justice to the rest of the tripitaka or to later (e.g. Mahayana) authors, which are unlike the suttas both in content and in style, so beware of that.
I recommend the answers to What should you do when someone teaches false Dharma? which help to answer this question.
You mentioned examples of "obviously non-Buddhist" and "seemingly Buddhist" teachings. I like to end this answer with a reference to AN 11.12:
"Furthermore, there is the case where you recollect the Dhamma: 'The Dhamma is well-expounded by the Blessed One, to be seen here & now, timeless, inviting verification, pertinent, to be realized by the wise for themselves.' At any time when a disciple of the noble ones is recollecting the Dhamma, his mind is not overcome with passion, not overcome with aversion, not overcome with delusion. His mind heads straight, based on the Dhamma. And when the mind is headed straight, the disciple of the noble ones gains a sense of the goal, gains a sense of the Dhamma, gains joy connected with the Dhamma. In one who is joyful, rapture arises. In one who is rapturous, the body grows calm. One whose body is calmed experiences ease. In one at ease, the mind becomes concentrated.
"Of one who does this, Mahanama, it is said: 'Among those who are out of tune, the disciple of the noble ones dwells in tune; among those who are malicious, he dwells without malice; having attained the stream of Dhamma, he develops the recollection of the Dhamma.'
I expect there are Dhamma-talks on this subject, but it's another way to distinguish true and false Dhamma; this is summarized again here:
The Six qualities of the Dhamma:
Svakkhato: The Dhamma is not a speculative philosophy, but is the Universal Law found through enlightenment and is preached precisely. Therefore it is Excellent in the beginning (Sila: Moral principles), Excellent in the middle (Samadhi: Concentration) and Excellent in the end (Panna: Wisdom),
Sanditthiko: The Dhamma is testable by practice and known by direct experience,
Akaliko: The Dhamma is able to bestow timeless and immediate results here and now, for which there is no need to wait until the future or next existence.
Ehipassiko: The Dhamma welcomes all beings to put it to the test and to experience it for themselves.
Opaneyiko: The Dhamma is capable of being entered upon and therefore it is worthy to be followed as a part of one's life.
Paccattam veditabbo vinnunhi: The Dhamma may be perfectly realized only by the noble disciples who have matured and enlightened enough in supreme wisdom.
I think there are some ("false") teaching which sound kind of baffling, aren't really actionable, don't seem to point in any specific direction.
I suspect there's something else important about any statement (dhamma or adhamma) you're considering i.e. the effect it has on you and how it may tend to affect interpersonal relationships.
Some point in a wrong direction: perhaps they are flattery for example, leading to conceit; or perhaps they're meant to make you dependent, by making you doubt any wisdom you have; etc.