The statement can be corrected by adding one word:
Theorem 8.7.1: Let $K$ be an ordered field. Then there is a complete ordered field $\tilde{K}$ and a dense order-embedding $\lambda:K \to \tilde{K}$ such that to each dense order-embedding $f:K \to L$ into a complete ordered field $L$ there is a unique order-embedding $f′:\tilde{K} \to L$ such that $f=f′\circ \lambda$.
Note that there are only two possibilities for a subfield $K$ of an ordered field $L$: either $K$ is dense in $L$ or $K$ is discrete in $L$. [If $K \cap (0,\varepsilon) = \varnothing$ then $K \cap (x-\varepsilon,x+\varepsilon) = \{x\}$ for each $x \in K$.] Being densely embeddable in a common larger ordered field is actually an equivalence relation on ordered fields. Cohn demonstrates that each such equivalence class has a unique maximal element up to isomorphism. I can't imagine any meaningful universal property that includes discrete embeddings.