How did conceptions of the Chinese tributary system differ between the political elite of China and those of the tributary states?

by fiahhu
EnclavedMicrostate

This won't answer every dimension of your question, but this answer by /u/thestoryteller69 discusses how tributary arrangements worked 'on the ground' in Southeast Asia. I would note, incidentally, that 'tributary system' is a deeply problematic term that captures very little of how tribute mechanisms worked and were conceptualised – see this answer for more detail.