--- Question ---
I have seen several definitions of 'additivity' in algebraic K-theory. In all cases, I can more or less see that there is something additive going on. But I have difficulty seeing how they intertwine. My question is: could someone explain in dummy language how these notions are related?
--- Definitions ---
Philosophical additivity. Algebraic K-theory is additive in the sense that it should convert short exact sequences into direct sums in some universal way. This is most easily seen in the definition of $K_0$ of a ring: a short exact sequence $0 \to M' \to M \to M''\to 0$ gets converted into a relation $M = M' \oplus M''$ in $K_0(R)$.
Additivity à la Waldhausen. Waldhausen has given an explicit additivity theorem for K-theory. He has several equivalent formulation, but to me the most lucid one is the following. If we have a cofibration sequence $F' \to F \to F''$ of functors $\mathcal{C} \to \mathcal{D}$, then $F' + F''$ and $F$ should be homotopic maps of spectra $K(\mathcal{C}) \to K(\mathcal{D})$.[1]
Additivity à la Blumberg et al. Regarded as a functor $\mathsf{Cat}_{\text{stable}} \to \mathsf{Spectra}$, algebraic K-theory should send split-exact sequences of stable $\infty$-categories to cofibre sequences of spectra.[2]
--- Why I care ---
I've heard that algebraic K-theory is somehow a 'universal' additive functor, as found in the paper by Blumberg et al., and the idea of a universal property of K-theory greatly entices me.
--- Sources ---
[1] : Waldhausen's Algebraic K-theory of spaces
[2] : Blumberg, Gepner, and Tabuada's A universal characterization of higher algebraic K-theory