Here's how I solved this problem for anyone who might have a similar one:
It seemed that the problem arose from the place where I installed the installation of the overpass ($ EXEC_DIR).
Therefore, I had to change the installation directories to:
$EXEC_DIR /var/www/osm/
$DB_DIR /var/www/osm/db/
$PLANET_FILE /var/www/osm/planet.osm.bz2
$REPLICATE_DIR /var/www/osm/rep/
default.conf:
<VirtualHost *:80>
ServerAdmin webmaster@localhost
ExtFilterDefine gzip mode=output cmd=/bin/gzip
DocumentRoot /root/osm-3s_v0.7.4/html
ScriptAlias /api/ /var/www/osm/cgi-bin/
<Directory "/var/www/osm/cgi-bin/">
AllowOverride None
Options +ExecCGI -MultiViews +SymLinksIfOwnerMatch
Order allow,deny
Allow from all
</Directory>
ErrorLog /var/log/apache2/error.log
LogLevel warn
CustomLog /var/log/apache2/access.log combined
</VirtualHost>
, . !