Hardness is defined by quantifying over NP rather than against a fixed complete problem. NP is available, so the definition a textbook gives can be written down.

The reduction's target is an instance with its threshold rather than a word, with the encoding applied by the definition. This keeps a statement about a problem from also being a statement about which words are well-formed.

"Strongly NP-hard on a class" is the same definition with one further clause, so that a single reduction witnesses both the plain claim and the claim on the slice the construction actually lands in — here, instances all of whose weights are one.

Strength is a clause on the reduction, not a different encoding. The usual formulation — NP-hardness under the unary encoding — says the same thing: a polynomial-time reduction writing numbers in unary is exactly a polynomial-time reduction whose numbers are polynomially bounded. Stating it as a bound keeps one encoding in play and makes the content visible, namely that the construction's numbers do not grow with the values it reads but only with the size of what it reads.

The polynomial is written as $(|x|+1)^c$ rather than as an element of $\mathbb{N}[X]$. Every polynomial with natural coefficients is dominated by such a power, so nothing is lost, and the bound stays elementary, in the style of the running times on the word RAM.