as Hume and Smith defined it

That people need other people is hardly news, but for Rousseau this dependence extended far beyond companionship or even love, into the very process of becoming human. Rousseau believed that people are not born but made, every individual a bundle of potentials whose realization requires the active involvement of` other people. Self-development is a social process, Self-sufficiency is an impossible fantasy. Much of the time Rousseau wished passionately that it were not: Robinson Crusoe was a favorite book, and he yearned to be free from the pains and uncertainties of social life. But his writings document with extraordinary clarity the shaping of the individual by his emotional attachments. "Our sweetest existence is relative and collective, and our true self is not entirely within us." And it is kindness- which Rousseau analyzed under the rubric of pitié, which translates as "pity" but is much closer to "sympathy" as Hume and Smith defined it - that is the key to this collective existence.

Source: a reading passage from an English examination

1. What is the reason that "as Hume and Smith defined it" could not be placed before "is"?
2. It would seem to me that "as" makes "as Hume and Smith defined it" a restrictive relative clause modifying "sympathy".
   What grammatical role does "as" play in the sentence?

This is a clause that describes "sympathy".

I understand that there are multiple definitions of "sympathy". Among these definitions is the definition of (David) Hume and (Adam) Smith, and you should understand "pitié" to mean the same as this definition of "sympathy", rather than the direct translation of "pity". The clause could not be placed before "is", because it should follow the noun that it modifies.

Rousseau analyzed kindness in terms of "pitié", which means "sympathy" as defined here:

By "sympathy" Hume means our capacity "to receive by communication [the] inclinations and sentiments [of others] however different from or contrary to our own." (source)