There are two basic kinds of potatoes: starchy and waxy.
The difference is actually in the amount of starch in them. Starchy potatoes, such as Russet or Idaho potatoes are best for baking, since the starch will allow it to be nice and fluffy after it's baked. They don't hold their shape particularly well when cooked. If you cut into a baked potato, it should be crumbly and dry.
Waxy potatoes include most fingerling potatoes, and most red potatoes. They hold their shape better when cooked, so they are good for soups and stews and potato salads.
Yellow-fleshed potatoes such as Yukon Gold or Yellow Finn are in between, and so can be used for either purpose. Purple potatoes also seem to have medium starch, and so can be used for both.
According to one cookbook I have (Moosewood Restaurant New Classics), there are two ways to determine which is which. If you cut the potato in half, the blade will have a film or a foamy residue on the knife. These potatoes may also stick to the knife because of the starch.
The rules for the brine is to make it of 11 parts water to 1 part salt. A high-starch potato will sink (it's denser) and a low-starch one will float.
Here's a good overview.
Now as to choosing ones that will bake well, it really depends on size. You want potatoes that will bake evenly, so try to choose ones of similar size. I bake my potatoes directly on the oven rack so that the hot air can move freely around them. Try to give them lots of room for that. Also, if you wrap them in aluminum foil, they'll steam and won't give you the nice crunchy skin. (And I'd avoid any potatoes with irregular shapes -- the smaller diameter ones may cook before the larger ones.)
Large potatoes take about an hour to bake at 350F. Lots of small ones may take a bit less time -- experiment with your oven and your potatoes. I'd start squeezing slightly after about 40-45 minutes. If they yield when squeezed, they're probably done.
It sounds to me like the issue may be that you're crowding the pan.
Basically, to get everything nice and brown and crispy, you need enough space for all of the steam to escape. That picture you showed has potatoes stacked on top of each other -- that means as the bottom items cook, they're going to end up steaming the items above them.
At a diner, they have a large griddle to work with -- they can really spread things out. You're not typically that lucky in a regular kitchen, as you don't have as much space, and you have a lip on the pans that'll hold the steam in.
So, either work in smaller batches, or consider recipes that use an oven -- using sheet pans instead of a pan on the stove solves much of the problem.
One other trick is that most diners don't start from raw potatoes -- maybe with hash browns, but not for home fries, you're not going to get the nice soft interior in a reasonable amount of time unless you start with a potato that's already been baked or boiled. (If you're doing things in the oven, you might be able to, but not in a pan)
Just for reference ... I have a 14" cast iron skillet that I use for home fries ... and it's about the right size for cooking a single large potato, which might be two servings, maybe three for kids. (I tend to cook carb-heavy meals).
update : I probably should've stated this directly -- you want the chunks of potato to form a single layer in the pan, with space in between them.
Best Answer
I'm assuming you're talking about shredded style hash browns. I'm not sure how much potato variety affects the outcome, as I tend to buy yellow potatoes (as they can typically be used either for roasting or mashing, so I don't have to think about how I'm going to use them in advance). I'm also more likely to make 'home fries' style hash browns, but I did a bit of research & testing for my pancake demonstration
Most recipes for this style call for soaking the shreds in water to remove the starch, then wring them out in a dish towel or similar. You can get sometimes get better adhesion if you don't soak them, but still wring out as much water out as you can.
If you're making latkes, you'll want a bit of depth to the oil, but generally 1 to 3 tablespoons of oil or butter is enough for hash browns, depending on how large of a surface you're working with.
Make sure the crust is nice and brown before flipping. Also, make sure that you're flipping it over in chunks suitable for your spatula size (either divide it up into quadrants or similar if coating the entire pan; or make individual piles only slightly larger than your spatula). If the potatoes drank up all of the oil when cooking the first side, it's generally a sign that the heat is too low and you might need to add some more (before you flip, so it has a chance to heat up)
The exact temperature to cook at depends on how much you're trying to cook at once, how fast your pans recover after putting the food in, and how thick of a layer you're cooking. (you want to make sure that the middle is cooked before you've burned the outside; if you're having problems with this, you can also par cook the potatoes in a microwave first).
If your pan/burner combination has a particularly slow temperature recovery, you might want to get the oil up to shimmering before you put in the shreds, but you typically aim for a little lower than that.