Let $K/F$ be an extension of number fields. We say that $K/F$ is 

 - a Weber-Hilbert class field if the prime ideals of $F$ that split 
  completely in $K$ are exactly the principal prime ideals.

 - a Takagi-Hilbert class field if the norms of all ideals from $K$
  are principal ideals in $F$: $N_{K/F} D_K \subseteq P_K$, where
  $D_K$ is the group of nonzero fractional ideals in $K$ and $P_K$ 
  the group of nonzero fractional ideals in $F$. 

 - an Artin-Hilbert class field if $K/F$ is abelian and if the Artin
  symbol induces an exact sequence 
  $$ 1 \longrightarrow P_K \longrightarrow D_K \longrightarrow 
     Gal(K/F) \longrightarrow 1 $$

Since these definitions involve ideals rather than ideles my guess is
that the most simple proof of the equivalence of these definitions
uses the classical approach. This approach in turn uses analytic tools,
and for these tools sets of prime ideals with density $0$ are invisible.
Thus in the classical theory, which is based on Takagi's definition, 
the equivalence with Weber's definition for all ideals (and not just
all except for a set of density $0$) is proved after all the main
theorems.

What is clear from the definitions is that if $K/F$ is a Takagi-Hilbert
class field, then each prime ideal that splits completely in $K/F$ must 
be principal. The proof that in a Takagi-Hilbert class field exactly the 
princiapal primes split completely up to a set of exceptions with 
density $0$ may be accomplished using analytic tools and should not 
be too difficult. 

It is also clear that an Artin-Hilbert class field is a Weber-Hilbert
class fields, since prime ideals have trivial Artin symbol if and only
if they split completely. For proving the converse you have to show that
a Weber-Hilbert class field is abelian. Arnold Scholz (<em> Zur 
Klassenk&ouml;rpertheorie auf Takagischer Grundlage</em>, Math. Z. 30
 (1929), 332--356) has shown that Takagi-Hilbert class fields must be
abelian, and perhaps this proof may be transferred to Weber-Hilbert
class fields. 

You will get a proof that Takagi-Hilbert class fields are Weber-Hilbert 
class fields by taking the classical approach to class field theory
based on Takagi's definition and following the proof up to the 
decomposition law, restricting everything to unramified extensions 
(which simplifies a few things, but not dramatically). If anyone
has a better idea, I'm all ears.