I am having trouble understanding the calculation of an idea, which is called 'cut height' in Negri and von Plato's Structural Proof Theory (SPT), and 'depth of a cut' in Troelstra and Schwichtenberg's Basic Proof Theory (BPT).
Cut height, according to SPT, is as follows:
And depth of a cut as defined in BPT:
As far as I understand, 'height' in SPT means the same thing as 'depth' in BPT. If we compare the cut height formulation to the (*) property, they also seem to be saying the same thing.
But looking at the example below from BPT, I would have expected that for the cut in the 2nd sequent calculus below to have a cut height/depth of $d_{01}+(d_{00}+d_{10})+1$. But BPT puts it as $\max(d_{00},d_{01})+1+d_{10}+1$.
So why the $\max(d_{00},d_{01})$? (And what is all the transformation he is doing there?) Am I mistaken in thinking that both SPT and BPT are talking about the same thing?
EDIT: Adding BPT's definition of depth below
The definition of cut-height in SPT (p. 35) is the same as the definition of level of a cut in BPT (p. 93).
Note that the notion cut-height in SPT refers to a size of an occurrence the cut rule in a derivation; whereas, in BPT, the notion of depth refers to a size of a whole derivation. Its definition in BPT is given in general as the depth of a tree, see p. 9. Since derivations are special cases of trees (whose nodes are sequents), this provides also a definition of depth of a derivation $\mathcal{D}$: it is the greatest number of successive applications of non-$0$-ary rules in a branch of $\mathcal{D}$ (so, it coincides with the definition of height of $\mathcal{D}$ in SPT, p. 30).
Even in the case of the proof of Theorem 4.1.5 in BPT (what you are interested in), the notions of depth of a derivation and cut-height of a cut do not coincide, even if you are considering only derivations whose last rule is a cut and you compare the depth of such derivations with the cut-height of such a (last) cut. Indeed, the cut-height of a cut $c$ is (in the terminology of BPT) the sum of the depths of the two derivations that are premises $c$, not their maximum (or their maximum plus $1$).
Concretely, concerning your case 3a in the proof of Theorem 4.1.5 in BPT, if $\mathcal{D}^*$ is the derivation \begin{align} \dfrac{\overset{\mathcal{D}_{01}}{\Gamma \Rightarrow \Delta, D_1} \qquad \dfrac{\overset{\mathcal{D}_{00}[D_1 \Rightarrow]}{\Gamma \Rightarrow \Delta, D_0} \qquad \overset{\mathcal{D}_{10}}{D_0, D_1, \Gamma \Rightarrow \Delta}} {D_1, \Gamma \Rightarrow \Delta}\text{cut}_\text{cs} }{\Gamma \Rightarrow \Delta}\text{cut}_\text{cs} \end{align} then the depth of $\mathcal{D}^*$ is $d^* = \max ({d_{01}}, \max (d_{00}, d_{10}) + 1) + 1$, as correctly stated on p. 97.