Confusion about "on behalf of"

I see this grating usage of on behalf of all the time though I believe the speakers/authors usually mean on the part of. Just this morning I’m reading the news and I spot it again, except that this time it’s coming from a professional source in a public legal context where you’d think they might be more likely to get it right and so I’m wondering if perhaps I’ve been wrong about myself all this time.

"For now, I will assume that based upon your comments, there is no
interest on behalf of either the City of Phoenix or Maricopa County in
furthering the past partnership (with) the Diamondbacks. Your candor
with respect to this issue will cause us to move forward in a
different direction."

http://www.azcentral.com/story/opinion/op-ed/ej-montini/2016/08/08/montini-arizona-diamondbacks-maricopa-county/88422616/

Is this actually correct usage? Any thoughts on the confusion? This reminds me of dependent vs dependency, comprise vs compose and even bring vs take. It’s certainly not unprecedented for people to be confused about asymmetrical relationships.

I don’t see this being connected to on behalf of and in behalf of but I throw this in for completeness:

https://en.m.wiktionary.org/wiki/on_behalf_of

And as evidence that I’m right, this source says they aren’t synonymous:

http://www.btb.termiumplus.gc.ca/tpv2guides/guides/wrtps/index-eng.html?lang=eng&lettr=indx_catlog_b&page=9v1hfkP9n1Rc.html

Sadly nothing in the quoted text strikes me as elegant prose in the first place so maybe my doubt is unfounded and this error is even more common than I realized.

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Edit

Based on the first answer below I think I need to add to my question. First I want to address why I’m not satified with Max Williams’ answer.

In my mind the city/county is capable of having interest. I think it’s pretty common to personify companies and governments and all kinds of things so I don’t think it’s ultimately significant that there’s an actual third party representative here in the middle of the conversation. I do suspect it’s part of the reason there is apparently confusion about the usage though.

Taking Max’s examples I think that grammatically those statements are different:

1a. She expressed interest on behalf of the city.

1b. On behalf of the city, she expressed interest. (Equivalent to 1a.)

…and…

2a. There is no interest on behalf of the City Of Phoenix.

2b. On behalf of the City Of Phoenix, there is no interest.

2c. Speaking on behalf of the City Of Phoenix, I can tell you there is no interest.

In the first sentences the phrase on behalf of the city is a modification of the verb expressed, or of the entire sentence depending on how you want to parse it. That’s how I interpret it anyway. In the second it clearly appears to be a modifier of the noun interest. I guess I’m contending that people have arrived at sentence 2a by working backward from 2c.

It’s not the interest that’s "on behalf" it’s her acting as agent in expressing it. When you attach the phrase to the wrong word it no longer makes literal sense to me though perhaps it has become idiomatic. Again I can see this as creating confusion for some people and possibly where the line started blurring.

My citation didn’t make it clear that the letter was directed to County Supervisor Andy Kunasek. As far as I know, as a county official he has no capacity to act as an agent of the city too. I noted the word assume in a comment. I guess in the backstory there’s probably no need to assume anything since Mr. Kunasek has already been pretty vocal and even later the author uses the word candor.

It’s clear to me that while the letter is directed immediately to Mr. Kunasek and the assumption based on his own words, the statement I will assume upon your comments, there is no interest on behalf of either the City of Phoenix or Maricopa County… cuts past his agency to speak the decision-makers themselves. As I’ve already pointed above interest on behalf of doesn’t make literal sense unless you contort yourself to make it fit. I don’t want to belabor this because my question was really not intended to be about this.

So let me cut through that complication and rephrase it. If I walk up to my friend today and say this it would be absurd to me:

I read a news article today and I’m really worried about us losing our baseball team. It said there is no interest on behalf of either the City of Phoenix or Maricopa County to work with the Diamondbacks organization and keep the team here. (Wrong)

I read a news article today and I’m really worried about us losing our baseball team. It said there is no interest on the part(s?) of the City of Phoenix or Maricopa County to work with the Diamondbacks organization and keep the team here. (Right)

Let me give you the more common scenario that I hear on the street. I hear it all the time but the article finally gave me a concrete example to cite.

Recently I speaking with a coworker. As it happens this lady is very prone to misuse of many other English expressions and it does at times make communication harder. We were talking about something related to a mistake that a client had made and while I don’t remember it exactly she referred to it as something like an "error on their behalf" when it was 100% clear that we were talking about blame — not any kind of Monopoly-esque bank error in their favor that made them the benefactors of a net positive.

Does anybody agree with that or am I entirely wrong?