Here is a proof that the ideal norm as defined in the books by Serre and Lang is equal to the ideal norm as defined in Swinnerton-Dyer's book. We will start from the definition given by Serre and Lang, state some of its properties, and use those to derive the formula as given by Swinnerton-Dyer.

Background: Let $A$ be a Dedekind domain with fraction field $K$, $L/K$ be a finite separable extension, and $B$ be the integral closure of $A$ in $L$. For any prime $\mathfrak P$ in $B$ we define ${\rm N}_{B/A}({\mathfrak P}) = \mathfrak p^f$, where $f = f({\mathfrak P}|{\mathfrak p})$ is the residue field degree of $\mathfrak P$ over $\mathfrak p$, and this norm function is extended to all nonzero ideals of $B$ by multiplicativity from its definition on (nonzero) primes in $B$.

Properties.

1) The map ${\rm N}_{B/A}$ is multiplcative (immediate from its definition).

2) Good behavior under localization: for any (nonzero) prime ${\mathfrak p}$ in $A$, ${\rm N}_{B/A}({\mathfrak b})A_{\mathfrak p} = {\rm N}_{B_{\mathfrak p}/A_{\mathfrak p}}({\mathfrak b}B_{\mathfrak p})$. Note that $A_{\mathfrak p}$ is a PID and $B_{\mathfrak p}$ is its integral closure in $L$; the ideal norm on the right side is defined by the definition above for Dedekind domains, but it's more easily *computable* because $B_{\mathfrak p}$ is a finite free $A_{\mathfrak p}$-module on account of $A_{\mathfrak p}$ being a PID and $L/K$ being separable. The proof of this good behavior under localization is omitted, but you should find it in books like those by Serre or Lang.

3) For nonzero $\beta$ in $B$, ${\rm N}_{B/A}(\beta{B}) = {\rm N}_{L/K}(\beta)A$, where the norm of $\beta$ on the right is the field-theoretic norm (determinant of multiplication by $\beta$ as a $K$-linear map on $L$). To prove this formula, it is enough to check both sides localize the same way for all (nonzero) primes $\mathfrak p$: ${\rm N}_{B_{\mathfrak p}/A_{\mathfrak p}}(\beta{B}_{\mathfrak p}) = N_{L/K}(\beta)A_{\mathfrak p}$ for all $\mathfrak p$. If you know how to prove over the integers that $[{\mathcal O}_F:\alpha{\mathcal O}_F] = |{\rm N}_{F/{\mathbf Q}}(\alpha)|$ for any number field $F$ then I hope the method you know can be adapted to the case of $B_{\mathfrak p}/A_{\mathfrak p}$, replacing ${\mathbf Z}$ with the PID $A_{\mathfrak p}$. That is all I have time to say now about explaining the equality after localizing.

Now we are ready to show ${\rm N}_{B/A}({\mathfrak b})$ equals the ideal in $A$ generated by all numbers ${\rm N}_{E/F}(\beta)$ as $\beta$ runs over $\mathfrak b$.

For any $\beta \in \mathfrak b$, we have $\beta{B} \subset \mathfrak b$, so ${\mathfrak b}|\beta{B}$. Since ${\rm N}_{B/A}$ is multiplicative, ${\rm N}_{B/A}({\mathfrak b})|{\rm N}_{E/F}(\beta)A$ as ideals in $A$. In particular, ${\rm N}_{E/F}(\beta) \in {\rm N}_{B/A}({\mathfrak b})$. Let $\mathfrak a$ be the ideal in $A$ generated by all numbers ${\rm N}_{E/F}(\beta)$, so we have shown $\mathfrak a \subset {\rm N}_{B/A}(\mathfrak b)$, or equivalently ${\rm N}_{B/A}(\mathfrak b)|\mathfrak a$. To prove this divisibility is an equality, pick any prime power ${\mathfrak p}^k$ dividing $\mathfrak a$. We will show ${\mathfrak p}^k$ divides ${\rm N}_{B/A}(\mathfrak b)$.

To prove ${\mathfrak p}^k$ divides ${\rm N}_{B/A}(\mathfrak b)$ when ${\mathfrak p}^k$ divides $\mathfrak a$, it suffices to look in the localization of $A$ at $\mathfrak p$ and prove ${\mathfrak p}^kA_{\mathfrak p}$ divides ${\rm N}_{B/A}(\mathfrak b)A_{\mathfrak p}$, which by the 2nd property of ideal norms is equal to ${\rm N}_{B_{\mathfrak p}/A_{\mathfrak p}}(\mathfrak b{B_{\mathfrak p}})$. Since $B_{\mathfrak p}$ is a PID, the ideal ${\mathfrak b}B_{\mathfrak p}$ is principal: let $x$ be a generator, and we can choose $x$ to come from $\mathfrak b$ itself. By the 3rd property of ideal norms, ${\rm N}_{B_{\mathfrak p}/A_{\mathfrak p}}(xB_{\mathfrak p}) = {\rm N}_{E/F}(x)A_{\mathfrak p}$.
Showing ${\mathfrak p}^kA_{\mathfrak p}$ divides ${\rm N}_{E/F}(x)A_{\mathfrak p}$ is the same as showing ${\rm N}_{E/F}(x) \in {\mathfrak p}^kA_{\mathfrak p}$. Since $x$ is in in $\mathfrak b$, ${\rm N}_{E/F}(x) \in \mathfrak a \subset {\mathfrak p}^k$, so ${\rm N}_{E/F}(x) \in {\mathfrak p}^kA_{\mathfrak p}$. QED