For a slightly different way to do it that is maybe more maintainable you could:

use strict;
use warnings;

my @lines = <DATA>;
my @records = ospf2_split_database (@lines);

print @$_ for @records;


sub ospf2_split_database {
    my @ospf2database = @_;
    my %data = (Router => [], Network => [], Extern => []);
    my $keys = join '|', keys %data;
    my $array;

    for my $line (@ospf2database) {
        $array = $data{$1} if $line =~ /^($keys)/;
        next unless defined $array;
        push @$array, $line;
    }

    return values %data;
}


__DATA__
Heading stuff that should be ignored

Router: 1st router line
2nd router line
this router line includes keys Network and Extern

Network: 1st network record

Extern: 1st extern record

Extern: 2nd extern record
Router: 2nd router record
[download]

Prints:

Network: 1st network record

Router: 1st router line
2nd router line
this router line includes keys Network and Extern

Router: 2nd router record
Extern: 1st extern record

Extern: 2nd extern record
[download]