Hi Stevie
It is simple:
"Install" has in the pc world the signification of «make a software resident on the actual pc».
At Linux, it has a second signification being «make a (Linux) distribution resident according to the rules of the distribution resident on the actual PC».
Why the difference?
Because the name Linux also has two significations 
. Linux is the name of the most important part of the distribution, the Linux kernel. It is not only valid for Linux distribution but also for Android for ex. Android also uses Linux, the same Linux kernel.
But under "Linux", we understand generally more: a distribution according the rules of the mother system "Unix" cloned as Linux distribution and having done some adaptation step by step as for ex. new hardware did appear. To clone Unix was the main goal of Linux as his creator did begin. For this reason, Linux did have (about) immediately a complete set of manuals explaining how that distribution has to work (completely different Android does not follow those rules although it uses the same main part, the kernel).
The distribution is an highly important point of view in Linux.
It is (now) possible to realize a Linux distribution with only a (very big) Linx (kernel), and all Linux is contained in it. And it is possible and was a long time necessary to add some functionalities required by the rules to really be a clone or successor of Unix by saving those supplementary active parts and settings to reproduce the real Unix system on a poor PC. Why? Because Unix was NOT initially a operating system for poor IBM PC's but a system for more important central computers used with one or more terminals and multi users as well as multi tasks. poor PC was not directly able to do that. but with the use of PC's did the number of environments and soon of new needs increase terribly. A professional Unix environment was terribly strong compared with the beautiful one-man IBM-PC and clones environment! And soon did an important difference of use appear, for ex. between a network station for a lot of terminals (not possible on the initial IBM-PC) and the office environments of IBM-PC's able to solve easily simple and beautiful jobs where central computers under Unix and more did have great difficulties:
The "distributor", the 2 older ones still active are slackware and suse, as well as a few later Redhat and Debian, was determinant to get a "Linux in harmony" with the goals of Unix (you must see that one of the most small Linux distributions able to be work in graphic mode, most Unix work station were not able to work in graphic mode, you can yet get actually did be contained on only 2 floppy disks, one full with only 1,5 Mo and the second half full with some applications for the graphic mode!
And that did also work (on IBM PC really arm an RAM): I was absolutely needing to subdivide the all distribution and pre-install all the part in an order so that after start the complex high security for a multi-users and multi-tasks environment in a network is present as under real Unix on a big computer, this also the PC station was a dwarf compared with an Unix computer...
This was the evolution since the publication of the first Linux (kernel) and the end of the 20th Century.
The Linux (distributions) did reach and offer more than Unix itself because the PC power did increase rapidly:
Linux (kernel) did become only a part in a lot of Linux (distributions) having each own rules but all analog to Unix: it was an obligatory condition to install each distribution according it's own rules with no deviation. And it was also a need to do that on a mass memory like hard disk, or later floppy disk, more later CD's and finally DVD's and USB-drives or mass memories.
For this reason the phase "installation" was very important, a major step or sometimes some taboo for guru's!
And the full installation is
that!
The next step as the size of distributions did increase faster as the memories and drives in the first years was as big memories and drives did appear internal and external on the market.
To simplify that, some freaks did try to adapt the Operating System to the size of at this time available memories. I did meet Barry Kauler at the black board of the distro "Menuet": the first Puppy's did reflect those Menuet's releases. Menuet was a try to shrink the Operating System by compiling it especially for one processor directly in assembler. That small distro was able not to exceed 1.5 Mo in 2000...2002 and offer a graphic environment like small Linux distributions like Basic Linux (baslin designed for DOS/Win95 PC's) but needing more than one FD to be complete (in Basic Linux, it was possible to draw with the mouse!).
Soon after this point did Puppy Linux appear as ISO able to be burn by each on a simple CD.
The new idea was:
Copy the very small content of CD into the RAM one time the day (or let the PC on all the time) and start the operating system itself immediately after the copy process in the RAM.
The design of the ISO file did have to contain all supplementary parts of initially jobs to do that, and, if not enough RAM, to try to start although that difficulty of leak of memory RAM enough of the system to permit it's use and hold on demand the rest again from CD (depending of the situation, the CD drive did become free or not for other uses, for ex. to listen music of course!).
This means: No installation any more at user side - Barry Kauler did compose the operating system, pre install it, also so, the PC can work needing only an very low size of RAM and a way to read the ISO.
Puppy was probably the first system really created to make automatically the choice (a few later, I did make very big remasterised copy of SliTaz as SliTaz did offer very comparable figures; but the size of the new ISO's compared with the size of available RAM was definitively a K.O. condition. I can't remember that SliTaz did come into the situation to maintain the CD in the drive to permit to continue to use the remastered ISO a certain time).
The sensational particularity of Puppy was born and with it Puppy as a great distribution able to be used without installation at user side, for this reason named frugal: after start through vmlinuz and initrd, Puppy uses only the pre installed and packed distribution in initially one file, today some squashed files (those with the extension *.sfs, this means "s"quashed "f"iles "s"ystem):
The Puppy (Linux distribution) is present on the PC about as a big Linux (kernel) containing all the Puppy distribution would be
or really as kernel-only-distribution (Barry Kauler did effective realize also that: all Puppy is in the very big kernel of some releases of Puppy = absolutely no installation at the side of user at the side of Linux: only the boot loader hat to be adapted. Puppy is in this case only one unique kernel file).
If you install full after that, you demolish some kind the specific propriety of Puppy to be completely pre installed 
