Are there differences in the ways/rationales that movies cast real persons to play "themselves?"

Hiring someone to "play themselves" is a very specific thing.

For example, in Ocean's Twelve, Bruce Willis literally plays Bruce Willis. That's the character's name.

Similarly, news anchors often appear as themselves in films set in the real world... Larry King has done it many times.

This is different than what I think you're talking about... which is called, in the industry, Real People Casting. The decision to go this route is made by the director of the film, not the casting director, though the casting director may recommend it if the role/s warrants it. Generally, casting directors will cast all actors unless the character has very specific need that can't be filled by actors. Remember, the casting director only acts as a filter and a source of advice for casting. He/She doesn't generally make any casting decisions.

When you follow this method of casting, you avoid casting actors in the scripted roles and, instead, find people who actually resemble your characters to some specific degree... perhaps they do the work your character does, or they have dealt with the same things in their lives. The person is still playing a character. The character doesn't have their own name, for example, and the director is still often telling the person how to perform a scene to get the performance they want.

One of the people who I think is best known for using this method is Terrence Malick. Many of the characters in Tree of Life are real people cast because he liked the way they look or who they are and roles are often created for them.

For The Tree of Life, Malick sought out non-actors for most of the characters and gave very little direction (as is his way). In his view, the worst thing you can do is cast someone who is naturally perfect for a particular role and then tell him or her to be someone else.

The preliminary casting for the main character in the upcoming Mike White pilot for HBO, Mamma Dallas, a comedy show about a drag queen, included outreach to real drag queens, though they eventually cast an actor for that role.

Real people can also be cast for technical reasons, for example, hiring an actual nurse to play a nurse. In Parkland at least two of the nurses in the OR scenes are actual nurses. They're not credited as actors, though.

Some casting directors even specialize in this type of casting (Google search). Generally, for safety, you're going to see some actors in addition to the real people. There are natural actors out there but, particularly for something like a waitress, I'm sure many actors will attest to the fact that they've had that real experience themselves.

There's also a couple of other cases that may apply:

A role being based on a real person:

This is common in all forms of fictional writing and is not limited to scriptwriting. In this case, the role is generally based on someone the writer (or writer/director) uses someone they know as inspiration for a character. The role is often still filled by an actor, either because the real person isn't interested in playing the role or isn't alive (or an appropriate age) any more... or just isn't good at acting. Even if you're just telling a person to be themselves, it doesn't always come across well on camera. That doesn't mean you can't consider that person for the role (or use real people casting to find someone similar).

Also, when you're basing a character on a person, you're creating a character based on your interpretation of the outward characteristics and personality of that person... and they may not even be aware of what they're doing, so if you ask them to do what they usually do, it may end up being different than what you remember.

This is difficult to document without some insider knowledge as the screenwriter doesn't usually have any role in casting decisions unless they're also the director or producer of the project.

Role being written for a specific person:

I don't think this really applies but I figured I'd mention it. Some writers (more often writer/directors) will create a role with a particular person, usually an actor, in mind for that role. This isn't uncommon but even with famous directors, they don't always end up getting that person to fill the role.