So, it seems we have a failure to communicate, perhaps. A HOME directory is a specific thing for each username/login. It is not system-wide.
echo $HOME
will show the HOME for the current userid. That is where individual settings and data are stored.
Where software gets installed depends on the type of packaging, but it is NOT in any HOME directory. Debian/APT packages almost always are installed under /usr/, with system-wide configuration files placed into /etc/. Before snap packages were recently forced onto Ubuntu users, this was always the way it worked. I think snap packages are stored in /var/ somewhere, but really don't know.
In general, / (the root directory) would hold the OS and application files. 25G was the recommended size for most desktop users. It you are cautious, you can fit a desktop in 10G or 15G, but that takes work, mostly by not installing lots of GUI programs. 14GB is small for many users. I've been moving the same desktop forward since the beginning of Ubuntu and it fits into 17GB - that's everything except the swap.
There isn't any way to tell new install programs to be installed outside /usr/. When that partition gets full, add a new partition+file system with the necessary space for all the old files AND all the new files and copy/move all the files over. This is best performed using a Try Ubuntu boot flash drive. In reality, as long as the files "appear" to be in the correct directory, with the correct owner, group, permissions and ACLs, where they are actually stored doesn't matter.
I think /home/ should be about 50G per userid, assuming they are full desktop users. Media files should be stored elsewhere, IMHO, but snap packages don't like when files to be accessed are outside the HOME.
I cannot provide guidance about snap packages. I'm not prepared to deal with those at all, so that entire subsystem is purged from my machines.
I suppose the next step you might take is to figure out which directories are using most of the 14G of space on /.
Or
if your install is using a swapfile - probably in /, then you can disable that, create a swap partition almost anywhere else, and free up the storage the /swapfile was using. That would be my first hope for solving something like this. I'm not a fan of swapfiles.
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