I see where I went wrong.

I checked the secondary files against the primary file (analyse subroutine), rather than the other way around. Easy enough to switch that:

use Modern::Perl;

use Number::Range;
use Text::CSV::Auto qw( process_csv );
use Data::Dump qw /dump/;

my $debug = 0;
my %database;
process_csv('./primary.txt',
    sub {
        my $row = shift;
        push @{$database{$row->{name}}}, [$row->{start}, $row->{end}];
    }
);
say dump(\%database) if $debug;

analyse ('./secondary_1.txt');
analyse ('./secondary_2.txt');

sub analyse {
    my $file = shift;
    say "--Checking $file--";
    my %data;
    process_csv($file,
        sub {
            my $row = shift;
            push @{$data{$row->{name}}}, [$row->{start}, $row->{end}];
        }
    );
    say dump(\%data) if $debug;
    for my $name (sort keys %database) {
        unless ($data{$name}) {
            for my $range_ref (@{$database{$name}}) {
                say "$name: $range_ref->[0] $range_ref->[1]  absent 0 
+0";
            }
            next;
        }
        for my $range_ref (@{$database{$name}}) {
            print "$name: $range_ref->[0] $range_ref->[1] ";
            my $range = Number::Range->new($range_ref->[0] .. $range_r
+ef->[1]);
            for my $testrange_ref (@{$data{$name}}) {
                if ($range->inrange(@$testrange_ref)) {
                    print "present $testrange_ref->[0] $testrange_ref-
+>[1] ";
                }
                else {
                    print "absent 0 0 ";
                }
            }
            print "\n";
        }
    }
    say '------------------------------';
}
[download]

Which produces the following output:

--Checking ./secondary_1.txt--
Alex: 3 44 absent 0 0 absent 0 0 
Alex: 124 175 absent 0 0 present 134 155 
Barry: 2 44 present 12 24 
Drew: 9 43  absent 0 0
James: 6 45 absent 0 0 
------------------------------
--Checking ./secondary_2.txt--
Alex: 3 44 absent 0 0 
Alex: 124 175 present 154 174 
Barry: 2 44  absent 0 0
Drew: 9 43 present 19 54 absent 0 0 
James: 6 45 present 29 45 
------------------------------
[download]

The whole range checking is done thanks to the Number::Range module. Please, check out its documentation and more specifically the inrange-method called in scalar context. I hand it the begin and endpoints of the range to be checked and if both are within the range to be checked against, it returns true, hence the whole range must be within.

The analyse subroutine in the script above takes as its only argument the filename of a "secondary" file and the primary file is checked against this secondary file. There is no secondary-against-secondary checking done or a reciprocal secondary against primary. Check the main for-loop inside this subroutine: it uses the data in the %database-hash which has been populated with the data from the primary file.