= Solution
A <first-order theory> is a <set> of <first-order sentences> in a fixed <first-order language>, often identified with its deductive closure. It is $\kappa$-categorical if it has a <first-order model> of <cardinality> $\kappa$ and all <first-order models> of that <cardinality> are <isomorphic> as <first-order structures> in that same <first-order language>. In particular, an <isomorphism> must preserve the named constants as well as the order.
A <first-order theory> is complete if it is <first-order consistent> and decides every <first-order sentence>: for each $\sigma$ it entails either $\sigma$ or $\neg\sigma$. Equivalently all its <first-order models> satisfy exactly the same <first-order sentences>. <First-order consistency> is included to exclude the vacuous <first-order theory> with no <first-order models>. The distinctions matter here: naming infinitely many ordered points changes <countable> categoricity without destroying completeness.
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