[fChk]

The Kernel Version of the OpenVZ VPS is:

Linux version 2.6.32-042stab145.3

Virmach KVM VPS that upgraded so effortlessly to Debian 10 was: 4.19.0-16-amd64

The VirMach upgrade was successful because you were using a KVM VM/VPS in it. KVM VMs have their own kernel that you can do whatever you like with them inside that VM (upgrade, downgrade, patch etc..) BUT an OpenVZ VM/VPS doesn't have a kernel of its own but it's using the Host's kernel. Thus kernel version 2.6.32-042stab145.3 is your OVZ-VPS host's kernel which you don't have permission to change.

Always Remember, OpenVZ VPSs aren't real VMs but CONTAINERS, just like Docker containers in a way, but functionally similar to LWD/LXC containers!

The system asks for a Kernel upgrade to minimum 3.1.  Am I correct that one can't upgrade a Kernel at OS level on an OpenVZ Virtualization?  Only sponsor can?  Whereas with KVM one can upgrade a Kernel in OS?

Indeed!.. for the reasons outlined above.

This is why:

@Mashiro  I've been trying to do the upgrade from Debian 8 to 10 on an OpenVZ VPS with 1 GB RAM and 60 GB SSD space.  For the first time I found a hickup with upgrading to Debian 10 - looks like the easy upgrades may be better with KVM?  I first tried the direct upgrade from Debian 8 to 10. That didn't work well and it was searching for packages but wasn't clear about what was missing.

This is most probably because Debian 10 includes packages that need a higher kernel version that you will never get, just like @Mashiro's said here:

You cannot run Debian 10 on OpenVZ RHEL 6 (2.6.32 Kernel). It's not supported by OpenVZ at all. Attempting an upgrade to Debian 10 from previous version will always fail! Pretty much up-to-date and more modern Linux distribution IS NOT supported by this EOL OpenVZ version and will never be. Main reason being the kernel and the software versions requiring features that only newer kernels provide.

Then I tried it incrementally.  From Debian 8 to Debian 9 first, and then Debian 9 to 10. The upgrade went OK (as far as I could see) from Debian 8 to 9.

In the sense that Debian packages upgraded to the ones shipped with Debian 9, yes!... Except the kernel! since you'll be still using the host's kernel, ie kernel 2.6.32-042stab145.3, whatever Debian version you'll be running.


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PS: This incident of yours reminds me of my first encounter with the OVZ-VPS @GigaRank :-) I wasn't aware at that time that OpenVZ VM were containers, till I learned it the hard-way!.. I think I touched on this in the review I've posted there.
There is always a first time for everything :-)