To me, Emerson's idea of self-reliance is very different from that which is commonly used. I think that we usually use this phrase to mean independence -- to mean that a person does not need other people to support them, especially economically. This is not what Emerson is talking about.
To Emerson, self-reliance is more of a moral idea. What he means by this phrase is that people should rely on their own ideas about what is wrong and what is right. He believes that people should not allow society to tell them what to do or how to feel. Instead, people should follow the dictates of their own consciences.
To me, these are very different usages. Our common usage is more mundane and every day because it is based on issues of economics and material things. By contrast, Emerson's usage is centered on morals and ideals, not material things.
I think that it has gotten to a point where one can only think of Emerson with the term "self reliance." The basic meaning of it in his work and now is to take the idea of "reliance on self" and actualize it. I think that this is consistent with Emerson's ideas. The notion of being distinct from society, allowing emotional frames of reference to guide individual actions, and establishing one's own sense of self as opposed to surrendering to conformist realities are consistent with both his use of the concept and today's vision. Where there might be difference is how the term and its conjoining "self help" have become an industry. It seems that everyone is talking about "self help" and finding their own voice. Here would be the dilemma: If everyone is seeking to be an individual, then what does it mean to be an individual? Is someone an individual because they see everyone else becoming one? I think that Emerson might have a difficult time trying to reconcile that the idea of relying on one's self and creating an understanding of one's identity is a process that "everyone" cannot be doing at the same time. At some level, I think that there might be some divergence from Emerson's vision and the modern incarnation of it.