Hallo, @carbon_dragon :- to the 'kennels'...
There are essentially TWO kinds of Linux user:-
The 'refugee', who has no interest in WHY Linux works the way it does, and who just wants to be able to install an OS, as quickly & simply as possible, and get on and use it to do stuff.
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It has to be said that the concept of a 'frugally installed' operating system makes very little sense to anyone who has only ever known & used "fully installed' ones. But t'other Mike has pretty much covered the basics.
A 'fully installed" OS requires a partition all to itself. It copies its files to that partition, then the /home directory - where 'user' directories go - can then expand up to the size of that partition, as things are installed.
A 'frugally' installed' Puppy can confine all its files to a single directory. Puppy's system files are contained within 2 or 3 highly-compressed, read-only files. At boot, the initrd creates a 'virtual' file-system in RAM, after which it decompresses & copies the contents of those read-only files into the virtual file-system. The end result, so far as the average user is concerned, is a standard Linux set-up with a full "/" file-system.
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As Mike said, the 'frugal' concept was originally developed to allow easy installation of a lightweight Linux distro actually inside a Windows install. In this instance, the Puppy will always create what's known as a save-file.....a kind of 'container' which contains a pre-formatted Linux file-system within it. This allows Puppy to function correctly on an NTFS file-system.
As it turns out, you can extend this concept to run multiple Puppies on a single partition, by the simple expedient of installing each one to a uniquely-named directory, all beside each other on the same partition. Puppy's boot-loader will then search two layers 'deep' to find a bootable kernel.
I would advise performing a 'frugal' install if you can, because most Puppy software is coded to work best with that manner of install. As stated further back in this thread, a 'frugal' Puppy is still very much the complete OS; 'frugal' merely refers to the fact that when installed, your Puppy takes up very little space on the disk when compared to any other 'lightweight' distro.
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As for vi; if that is what you want, it's simple enough to install as soon as your Puppy is up-and-running. That won't be a problem.
Our admin, rockedge, has given you a good outline, which I, personally, still think is the easiest way of installing Puppy; burn the ISO to CD, boot from that CD, then install from within the running Puppy. (This is why, although my current rig is less than 2 years old, I went out of my way to find one of those rare & almost 'mythical' beasts that still come supplied with a built-in optical drive.)
I understand your concerns about the installer itself. As it stands, the 'Ubiquity' installer, used by more distros than any other, DOES tend to 'hand-hold' you through the procedure, largely automating most of the stuff that gives folks the biggest cause for concern....like formatting partitions, etc. Puppy, it must be understood, is mainly a 'hobbyist' distro, for folks who LIKE 'tinkering' and doing things for themselves. Consequently, things like formatting with gParted does tend to be a 'manual' procedure, to be undertaken by the user themselves.
It's not ideal, though the community has been working to address these kinds of issues in recent years. When you DO understand exactly what's happening, and why things do what they do (and why!), it's perfectly possible to perform a 'quick & dirty' install, of ANY Puppy, within 5 minutes or so.....
Any further questions, don't be a stranger..!
Mike.