Nice cleanup. I went through each function and compared implementations, and there are some things that are missing or inconsistent. Given the fixes that have gone into inlineEditor over time, I'd like to make sure they're as consistent as possible. Made notes of what I saw.

"Backbone" and "JavaScript" in the summary/description.

Can this live in Djblets? Seems a more natural place for it, given that's where the prior version lives. I'd also eventually like to use this in Splat.

Class is missing a doc comment.

This is missing support for useEditIconOnly and the logic around it.

Backbone.View.prototype.initialize is guaranteed to be a no-op. We don't need to call it.

Looks like an old type.

Let's do .hide() first. It'll save some churn.

Let's .hide() first here too.

Instead of an extra function, can we pass this.submit.bind(this) and this.cancel.bind(this)?

The key property doesn't look completely supported in Edge and appears completely unsupported in Safari for these keys. Can we go back to using keyCode and the constants for now?

/** -> /*

Blank line between these.

Can we use .bind(this)?

Can we take options instead, for future enhancement?

Same here.

This is referring to outerHeight itself and not calling it, and isn't accessing the option correctly. Needs manual testing.

This no longer recalculates the dirty state in this method. Was that not needed?

Logic here is different from what we have in the original code. This isn't doing the throttle anymore, which was added intentionally. (It was also doing a _.defer(), but this was prior to the _.throttle(), and may not be needed).
This should also go below public methods.

This should go below public methods.

Original code normalized initialValue first. The new code is doing it in the default isFieldDirty, but TextEditorView doesn't benefit from that.

Can we call this isDirty()?

Can you compare to 0?

Missing docs.

The original version also set $el.data('text-editor'), which was accessed by other things. Might be worth adding that back?

buttons used to be public, and now it's private. Can we go ahead and make this $buttons?

Instead of accessing a private variable, can we make $field public?

Some minor suggestions, this all looks pretty great :)

Can we call it animationSpeedMS? The abbreviation doesn't save much.

Since this isn't a closure over variables, we can pull it out into a function and just pass it in instead of recreating it every time we make a new InlineEditorView.

Should this have aria-hidden="true" since its a visual cue only and we don't want screen-readers to attempt to read the button?

Any reason we don't set this up in this.events ? Or in delegateEvents ?

Should we pul this out into this.$window in initialize?

Might as well make this one line.

This can be one line (no need for {})

You can use this._updateDirtyState.bind(this) instead, remove an unnecessary anonymous function.

Ship It!