Your partitioning looks OK.
If this computer has UEFI bios.
That 1st partition may also need flagged boot, esp
Needing a swap partition depends on how much RAM the computer has.
With Puppy Linux, if there is 4GB or more RAM the swap will probably not get used.
But it is good to have if it is ever needed.
Do you have some version of Puppy installed on a USB stick or a CD that you can boot the computer with?
If no.
You need to do that and boot the computer to this running the computer.
Run the installer programs in it.
Have the Slacko 8.2.1 ISO file so you can use it for the source the installer program will use.
Frugalpup Installer program will be able to do the install to the internal drive.
Maybe this info for using Frugalpup Installer will help you.
Drive labels are going to depend on number of drives on the computer.
So, you do need to be careful to select the correct drive label, when offered to select.
This is what I do.
Use Gparted program.
Setup the drive with 2 partitions.
First one, small 300MB, fat32 format, flagged boot. (location for boot loader files, boot partition)
Rest of drive, whatever other partition(s), but one ext3 or 4 format. (location to put frugal installs)
(this is UEFI standard requirement, and some computers, look for a fat32 partition, for boot loader files)
Run Frugalpup Installer main program.
On the main window are selection buttons.
Select the Puppy button, to do the install.
Go through install process, selecting to install to the ext formatted partition.
Note:
When selecting the partition to install to.
A window pops up, giving option to make a directory, to put the frugal install in.
I make this directory and usually name it, the name of the Puppy version.
Carefully read that windows info.
Press enter, makes the directory, not the OK button.
complete the install.
When it gets back to the main Frugalpup window.
Select the boot button.
Select the location of the frugal install, on the drive.
Select the small 300MB 1st partition as location to install the boot loader. (may need to scroll the selection window)
Select the boot loader type.
UEFI
mbr ->legacy bios boot
both
You can install the UEFI for UEFI computers, mbr for legacy bios computers, or both, to boot anything.
The UEFI will also install the needed files, to support secure boot enabled in UEFI.
When you first boot the drive, on a UEFI computer, with secure boot enabled.
A process will start, to allow you to install the Puppy security key, to the computer.
It will add this Puppy key, to the other ones, loaded on the computer.
To put more frugal installs on the same drive.
Do the complete process again, for the new Puppy version.
When you run the boot loader install.
It will make entries, for all installs it finds, on the drive.