= Solution
In a <countable> <first-order model>, the finitely many named constants split the order into the open intervals
$$
(-\infty,c_1),\ (c_1,c_2),\ldots,(c_{n-1},c_n),\ (c_n,\infty),
$$
together with the $n$ named points. Each interval is a <countable> <dense linear order without endpoints> in its induced order. It is nonempty and infinite by density and the absence of endpoints.
Use Cantor's classification of <countable> <dense linear orders without endpoints>: each such interval is <isomorphic> to the rational order. It follows by a <back-and-forth method> extending finite increasing partial <bijections> alternately along enumerations of the two orders. Given two <first-order models> of $\mathrm{DLO}_n$, choose <isomorphisms> between their corresponding intervals and send each named point to its counterpart. Their <union> is an increasing <bijection> preserving all constants, hence an <isomorphism> in $L_n$.
There are <countable> <first-order models>, for example any finite increasing list in the rational order. Therefore
$$
\boxed{\mathrm{DLO}_n\text{ is }\aleph_0\text{-categorical}.}
$$
For $n=0$, the same argument is just the classification theorem for the whole order.
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