What type of wood siding should I use for a shed

It probably depends on how much effort you're going to be putting into it.

If you're just planning on slapping something together quickly, then sure, go for untreated, and if something goes wrong, you can rebuild it in a couple of years.

I'd personally use pressure treated, as I'd have to make a larger structure due to local ordinances. (I can't store firewood within 18" of earth; I don't know if it's an attempt to deal with an invasive species that caused a ban on exporting firewood from our county) Because of the extra work involved, any savings on cheaper materials would be wasted if I have to rebuild.

I'd also be interested in the anti-insect properties of pressure treated, not only the anti-rot aspects.

The 8" Home Depot siding will be too narrow. It's 8" nominal, the actual exposure width will end up under 7". To avoid the need to exactly match widths, consider replacing the boards full height, then you have no butt joints in the siding.

If you don't want to spend for the extra material, finding siding that exactly matches width may not be that easy. You could get someone to mill boards to a custom width. Since you boards have no special exposed profile, someone with a table saw could easily cut boards to match what you have. The raw material would just be nominal 1x12 and 1x10 lumber with one face resawn, which is a pretty common siding raw material.

This is a relatively easy DIY project, but it may take more than 3 hours. Be sure the siding nails you use are either hot dipped galvanized or stainless steel. Electro-plated galvanized nails will still rust. Siding nails have more of a head than finishing nails, so the boards are held tight and can't warp, but smaller than regular nails so the exposed head is not too objectionable.