Minimal DFAs are unique up to start-state-preserving isomorphism

A minimal DFA has no unreachable states. Indeed, if $D_i$ had an unreachable state, then [L3] would remove it without changing the language, contradicting minimality.

Let $D_L$ be the canonical Nerode-class DFA from [L1]. Because step 1.1 shows that every state of $D_i$ is reachable, for each state $q$ of $D_i$ we may choose a word $u$ with ${\delta }_i^{\ast }(q_{0,i},u)=q$, which is possible by [L2], and send $q$ to the Nerode class $[u]$. If two words $u,v$ reach the same state $q$, then every continuation is accepted or rejected identically from that state, so $u$ and $v$ determine the same Nerode class. Thus the map is well defined. For any Nerode class $[u]$, the state reached by $D_i$ on $u$ maps to $[u]$, so the map is surjective onto the state set of $D_L$.

Both $D_i$ and $D_L$ are minimal DFAs for $L$: $D_i$ by hypothesis and $D_L$ by [L1]. Therefore their finite state sets have the same cardinality. The surjection from step 2.1 between finite sets of equal size is therefore a bijection.

The bijection of steps 2.1 and 3.1 sends the start state to $[\epsilon ]$, and its definition via reaching words shows that it preserves letter transitions and accepting states. Hence each $D_i$ is pointed-isomorphic to $D_L$, so $D_1$ and $D_2$ are pointed-isomorphic to one another.