In my use case, bring able to move my connection in another country (only possible with renting a VPS abroad in the aforementioned options) is not a necessity, but as it further hides my true location, that would be a plus.
Yes, this is exactly what a rented VPN will do for you.
Any connection you make to an outside entity through the VPN tunnel will appear to that service provider to them AS IF you are actually coming from the VPN endpoint’s country of origin instead of wherever you may be.
As far as your title question - the 101 of VPN - here’s my stab at it:
VPN’s are tunnels between two computers. Traffic across that tunnel is invisible to anyone but the VPN host and the VPN client. Corporate networks use this often to do business, banks, so on. Its of course encrypted with private key exchanges on each end, meaning even if the tunnel traffic were captured, its contents would be unknown, its just data.
In the case of a rented VPN, you pay a VPN provider to host and endpoint for you. NordVPN for example has hundreds of endpoints all around the planet - that is where their servers reside. When you make a connection to that endpoint from your VPN client, it is as I said earlier, it appears to any service provider on the other end to them as if your traffic is coming from the VPN provider’s servers rather than your personal computer or network.
In the case of a self-hosted VPN in a homeserver situation, this is often used so that you can remote into your homeserver when you are away from home via laptop, wifi, while at work and so on.
Otherwise, VPN’s are typically used in consumer scenarios with a VPN server being rented and you using a client to obscure your traffic.
As another commenter already mentioned: The majority of your traffic, especially web-based, is already encrypted with SSL/TLS/aka “https” anyway. However, your ISP, the website/service provider and everyone in-between know your IP address when doing that. Even moreso: SSL traffic is easily decryptable and many consider it unsafe due to man in the middle attacks. This often occurs on corporate networks, like at an office job: IT will decrypt SSL traffic real-time, and store the packets for later review, completely defeating the purpose of encryption entirely and violating your privacy. Free public wifi services at like McDonalds and Starbucks also do this.
All that being said, VPN is not an end-all-be-all solution to remaining private on the internet as tech media often suggests; it is however one major tool for such, or a major tool in the toolbox.
Other behaviors are required for OPSEC if you truly want to be private; this includes avoiding leaking your IP or identity accidently through other applications even while tunneling traffic through a VPN provider.
Hope this helps!