Why the term 'unified' in "unified mass unit"?
Your question is historically important. Nobody has the energy to write an answer which would require several pages of an article. In short, it is a can of worms. For almost 5-6 decades chemists and physicists used a different scale for the atomic masses, so the atomic masses in the publications/books used by physicists did not exactly match what the chemists used for their own publications. Note that it is just not a matter of academic egos, the entire chemical industry is dependent on atomic weights (for calculations and trade). Money matters. This is why both the governments and the scientific community wanted to sort this out.

Both parties were not willing to budge because that would have affected their millions of pages of research. However a compromise was reached to use a common scale by two three sensible men i.e., they proposed to choose C-12 in the 1960s instead of oxygen. This process sort of "unified" the community of physicists and chemists and it did not affect the values of either party by much.

More history is given here, The AME2003 atomic mass evaluation (I). Evaluation of input data, adjustment procedures, Nuclear Physics A 729 (2003) 129–336

There are scores of other articles from the 1960-70s on the choice of C-12. Use Google Scholar. The original work is in French I guess, I did not check

T.P. Kohman, J.H.E. Mattauch and A.H. Wapstra, J. de Chimie Physique 55 (1958)
393.