Backup Linux system to web
When it comes to system backups, I think it's more important to know what you include than to know what you exclude.
So instead of specifying what you exclude,
I think it's better to think about what you need to include.
It might be a shorter list,
and less noisy without all the --exclude flags.

Your current method is wasteful: the backup takes twice the size it needs,
once for the .tar.gz, and once for the split version. You could pipe the output of the tar command to split to save some space, for example:

sudo tar -cvpz --one-file-system /usr /etc /home ... | split -b 100MB - "/HDD/systemBackup/$date-"


Most importantly, as @tarleb also said, make sure to test restoring from backup. An all too common mistake is to not test restoring, and one day when you really need it, your backups turn out to be useless.


  Will this method also work on both a Raspberry Pi and an Ubuntu system?


The list of directories you want to backup might be a little bit different.
If you have to inspect both systems and see.
Other than that, the same technique should work, yes.


  Is this a good method of backing up my Linux system to the web?


This is one way to do it.
It's a full backup.
Its advantage is that it's simple.
Its drawback is that it wastes a lot of space,
because much of the data in each backup for different $date values will be duplicated.
And transferring over the network will be relatively slow.

An alternative would be using incremental backup,
which would save disk space and network bandwidth,
at the expense of considerably more complex backup and restore procedure.

One more way would be to rsync the interesting directories,
and take create a timestamped archive at the other side.
That would save disk space locally, save network bandwidth,
and still be relatively simple.