Discrete Wavelet Transform output: coefficients or FIR-filtered signals?
In one level DWT, each output of the low-pass or a high-pass can indeed be considered as signals. Thus each of those signals are subsampled by a factor of 2, and the same two-filter-subsampling is iterated on the low-pass output, several times (wavelet decomposition) at $L$ levels. Each final output of the different branches could still individually be considered as "signals", but they only make sense together, with respect to the input signal. Coefficients is a traditional name for them. Honestly, I don't know where it appeared originally.

Each represents information cast at a certain scale, and they don't really belong to the same domain. They are signals projected onto a nested wavelet subspace from a multiresolution analysis. They altogether belong to a global transformation. In other words: the values observed are signals with a heavy wavelet footprint (especially because of the subsampling). Analyzing them alone (without taking into account the wavelet projection) is theoretically dangerous, and even called "wavelet sin", or "wavelet crime", see for instance: Wavelet transform: How to compute the initial coefficients when only samples are available? (Strang and Nguyen, Wavelets and Filter banks, 1996, pages 232 sq.). However, people often may sin with practical efficiency, and for orthogonal non-redundant transformation, it often does not go too bad. 

Anyway, due to the invertibility of the DWT scheme, you would have "more genuine" signals by: keeping some coefficients (setting the others to zero), especially from each "subband coefficient set alone", and performs the (unique) inverse DWT for each of the  $L+1$ sets of subband coefficients. From those $L+1$ inverted, you get $L+1$ reconstructed signals in the original domains, whose sum yields the original signal (linearity of the wavelet).

The "wavelet crime" often becomes more stringent when one uses union of wavelet bases, wavelet frames, etc. As for the MODWT, I would not claim (yet) for sure that I fully understand the concept. Its setting sis akin to undecimated, shift-invariant cycle-spinning or redundant discrete wavelets. While the DWT generates $N$ coefficients for a $N$-sample signal, whatever the levels, the MODWT generates about $(L+1)N$ coefficients. As far as I get it, MODWTMRA does a job similar to the one described above for the DWT: project the redundant wavelet coefficients back to th original domain, so that their sum yields the original signal back. Side note: for redundant transforms, the inverse is not unique.