The precise meaning of 'coefficient' in signal processing when I read papers on signal processing, I often see the term 'coefficient' being used to refer directly to 'samples', i.e. "We can reduce the number of coefficients, by simply subsampling the even samples." or in presentations on the lifting scheme (wavelets), it seems the output of the low pass filter is also called coefficients. 

To me, the output of a low pass filter is a 'smoothed' version of the original input sequence. Further, to me, a coefficient is e.g. a, b in  y[n] = ax[n] + bx[n-1]. In this sense, the coefficients a and b certainly is a representation of the signal-- but i would not call y[n] the coefficients. 

I am probably very confused about the terminology. Could someone clarify this with some intuition?
There's really not much I can clarify for you here:

A coefficient really is just a factor in series. Whether that series is e.g. a Fourier series, a sum over powers of 2, filter delay taps or just a series of sampling instants can't be said. Context!