The computation-word convention agrees with the published finite-word definition

For each natural number $n$, both [L1] and [L2] say that a word of length $n$ over $\mathrm{\Sigma }$ is a function $n\to \mathrm{\Sigma }$. So the two conventions have exactly the same length-$n$ words.

Both [L1] and [L2] call the unique word of length $0$ the empty word $\epsilon$, so the two empty-word conventions coincide.

If $u:m\to \mathrm{\Sigma }$ and $v:n\to \mathrm{\Sigma }$, then [L1] defines $uv$ by taking the first $m$ values from $u$ and the next $n$ values from $v$. That is exactly the displayed word obtained by writing the letters of $u$ followed by the letters of $v$, which is what the published notation $uv$ of [L2] denotes.

Since the words of every length agree by step 1.1, their union over all lengths agrees as well. So the set ${\mathrm{\Sigma }}^{\ast }$ of Computation alphabets, words, the empty word, and ${\mathrm{\Sigma }}^{\ast }$ is literally the same set of finite words already used in Finite words, contiguous factors, avoidance and proper-prefix states.