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If I were doing it my choice to cut the boards in order of preference would range from :-
Track saw with depth stop ( or circular saw)
Multitool with depth stop (Fein Multimaster, other brands are available)
Jigsaw (It's doable but a bit erratic)
Hand Saw (japanese pull saw, use the correct blade type for sawing across the grain)For pulling the nails out you can use a pair of London pliers, you can pull the nail through the bottom of the board or take the out of the joist fairly easily with those.
A sharp wood chisel helps for neatening up the ends of the boards which remain.
Also, as well as that floorboard lifter it's handy to have either a small crowbar or flat utility bar (like a flat crowbar) for additional prying abilities. You can get that into more places and at more angles than you can the lifter. With a friend you can even apply pressure in multiple points at once which helps.
If the old boards are nailed down, you'll want a way of getting the nails out of either the joists or the boards (some pull up, some stay where they are). The most effective tools I found were a crowbar with a claw on the U-shaped end and a nail puller like this:
https://www.machinemart.co.uk/shop/product/details/irwin-vise-grip-8-nail-puller-pliers