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.
I keep a box of instant mashed potato flakes in the house. I use it only for two things: one is a super-quick pantry chowder and the other is to thicken over milked mashed potatoes. While I wouldn't want to eat a bowl of instant mash, a quick shake in this circumstance takes care of the problem and no-one's the wiser.
Best Answer
Here is how we made pomme puree at the restaurant I used to work at, for a very well known (in Canada anyway) French chef:
Peel and boil as many potatoes as you need. Cook until slightly underdone.
Run the potatoes through a ricer/food mill. Then--this is key--scrape the result through a tami--a very fine mesh sieve. Most often used for sifting flour. Usually about 14-18" diameter, looks like a drum.
Return resulting potato to a pot, add melted butter, cream, salt, pepper, nutmeg to your desired consistency, stirring all the time.
If not serving immediately, chill as fast as possible, and reheat to order in a pan using a little more cream to loosen it up.
IMPORTANT: you must work as fast as possible; the potatoes need to stay hot the whole time or will become gluey.