How exactly did Seneca Crane die?

We don't know, as the films/novel appear to differ.

In the films, it is initially implied that he ate the poisoned berries though. We see him locked in the room at the end of the films with the berries.

However, as you say Plutarch tells Katniss something different:

Plutarch: Seneca decided to quit breathing.
Katniss: Decided?
Plutarch: It was that or poison berries.

Later, when undergoing her assessment, she ties the noose around the neck of the dummy and lets it fall, which is where the hanging implication comes from.

However, in the novels this plays out slightly differently. We don't read anything about his death at the end of the first novel. Instead, we are told it at the start of the second, courtesy of Snow in Chapter 2:

“If the Head Gamemaker, Seneca Crane, had had any brains, he'd have blown you to dust right then. But he had an unfortunate sentimental streak. So here you are. Can you guess where he is?” he asks.

I nod because, by the way he says it, it's clear that Seneca Crane has been executed.

Another difference is her meeting with Plutarch:

“...So, you're the Head Gamemaker this year? That must be a big honor.”

“Between you and me, there weren't many takers for the job,” he says. “So much responsibility as to how the Games turn out.”

Yeah, the last guy's dead, I think. He must know about Seneca Crane, but he doesn't look the least bit concerned.

Plutarch flashes her a symbol of a mockingjay on his watch, but he never mentions Seneca's death.

She does, like in the films, drop a dummy with "SENECA" written on it (and she stains it with berries). Following this, she tells Effie and Cinna about this:

“I guess this is a bad time to mention I hung a dummy and painted Seneca Crane's name on it,” I say. This has the desired effect. After a moment of disbelief, all the disapproval in the room hits me like a ton of bricks.

“You ... hung ... Seneca Crane?” says Cinna.

“Yes. I was showing off my new knot-tying skills, and he somehow ended up at the end of the noose,” I say.

“Oh, Katniss,” says Effie in a hushed voice. “How do you even know about that?”

However, it's interesting to note that Effie doesn't actually say he was hanged. Her response seems to simply indicate surprise Katniss knows he is dead. The manner of his death isn't really discussed. Although Katniss stained him with berries, this seems more in response to her own actions with the berries, rather than any knowledge that that is how he was killed.

Conclusion:

The novels are particularly ambiguous about his death. We know he's gone. We have no canonical answer as to how though.

The films initially showed the berries as the cause of his death, but if Plutarch is to be believed, he killed himself some other way. What this way was is not known.