This truth is universal
The "secondary detail" mentioned by Vivekananda is what enables this universal teaching to be applied to different cultures in different times and circumstances. If you examine all religions closely, you discover that they all have the same purpose to manifest the hidden deity in man. The method is basically the same. Differences that appear only on the shallow aspect.
"Truth is one; only This is called by a different name. Everyone is looking for the same Truth; the variations are due to climate, temperament, and name. A lake has many ghats. From one ghat the Hindus take water in a jar and call it 'jal'. From another ghat Mussalmāns (Muslim) takes water in a leather bag and calls it 'pāni'. Of a third of Christians take the same thing and call it 'water'. Suppose someone says that it is not 'jal' but 'pāni', or that it is not 'pāni' but 'water', or that it is not 'water' but 'jal', It will indeed be ridiculous. But this is the root of friction between sects, their misunderstandings and quarrels. This is why people injure and kill each other, and shed blood, in the name of religion. But this is not good. Everyone goes to God.
It is important to have faith in its own spiritual path. It is natural and even helpful to believe that the way itself is the best. But one great message from Vedanta is to always remember that your own path is not the only way. All paths are correct. The proof is that people have realized God in innumerable ways. God is the only author of truth, and he can not be protected by anyone's copyright. Remember that in a sense, whether they do it consciously or unconsciously, everyone practices the Eternal Religion wrapped in different clothes.