The etymology is genuinely as described: a Sanskrit compound meaning something close to "salutation to you". What has drifted is the usage.
As a normal, slightly formal hello. You say it to a shopkeeper, a colleague's parents, a stranger. It is respectful and neutral — exactly what a learner needs, because it is never wrong.
What it is not is intimate. Friends do not greet each other with नमस्ते; they use a name, or hi, or nothing. Using it with a close friend sounds oddly ceremonious, in about the way "good day to you" would in English.
The phrase "the divine in me bows to the divine in you" is a yoga-studio elaboration, not the everyday meaning. It is not offensive, and it is not what a person in Delhi has in mind when they say it to the person selling them vegetables.
नमस्कार is a shade more formal and common in broadcasting. आदाब and सलाम are used in Muslim contexts, sat sri akal among Sikhs. Matching what the other person says is both the safe rule and the courteous one.
Reading an answer is one thing; being corrected while you say it is another. EkGuru tutors teach one to one from $6 a lesson, and most offer a trial.