I have not read all posts in this topic, so I may say something already posted.
I would take and start over with the internal drive setup.
Do this from a Running Puppy you can get booted from a USB stick.
Use Gparted program to completely setup the internal drive. (do you fully know how to operate Gparted?)
Make a new partition table on it (type msdos)
(that should completely delete everything on it, including any hidden partitions or files.)
Who knows what Lenovo cold have hidden on the drive.
Who knows what Lubuntu installer did to the partition table.
Now make the two partitions as you did before.
Small first partition formatted fat32 flagged boot and esp.
Rest of drive 2nd partition format ext4.
Install the Puppy versions as frugal installs on the 2nd partition.
Each one in it's own directory.
Now you need a boot loader installed to boot them.
If frugalpup Installer is available to use.
The boot loader it installs has option to install uefi, legacy bios, or both boot loaders.
So if unsure install both.
Frugalpup Installer
viewtopic.php?t=337
Install the boot loader to the first partition.
Computer bios setup settings.
Disable secure boot.
Look for anything like enable legacy boot, CSM, and if there, select it for how the bios operates.
Just to eliminate maybe an issue.
When trying to boot from internal drive. Do not have any USB drives plugged in.
If still not booting.
From the 1st partition.
Post the grub.cfg file contents so we can see the menu entries it is trying to use.
Note:
Frugal install of a Puppy version is still the complete OS, just installed in a special way.
A lot of Puppy features only work as a frugal install.