The fact that Twitter is a customer service channel, the tweet was sent with "Qualtrics Social Connect" which describes itself as a customer service tool, the fact that there was a personal sign-off on the tweet which big companies don't tend to do when they're making policy announcements, but that they do tend to do when a customer service rep is replying to a customer question. Plenty of large companies do this sort of support via their official account rather than through a dedicated support account, it generally provides a better customer experience because no one wants to look up an account like "LufthansaSupportEMEA".
I'm not sure what makes you think this is not a customer service interaction?
> the fact that Twitter is a customer service channel
This is circular reasoning. Nowhere is it established that Twitter is a customer service channel.
Other parts of your in your first paragraph makes sense, but are also non-obvious and reflect expertise and evidence collecting on your part.
Plenty of large companies do also use Twitter for PR rather than customer service.
Unless clearly stated otherwise, there is no reason for anyone to treat an official Twitter account as anything other than an official statement by a corporation, no different from their official blog, or website.
It sounds like you are coming from the position of "prove to me Twitter does customer support", and you're right I went and gathered some evidence for this. If you started from that position though I think you're entirely justified in not realising this was a customer support interaction.
However, in my experience of ~14 years on Twitter, seeing support processes inside companies, and as a consumer, I think most consumers treat any two-way comms channel from a company as customer support regardless of whether the company is providing it. Many even treat review websites as customer support, despite it being wholly inappropriate for that.
Companies do support on Twitter because their customers are treating it like that, not the other way around.
There's a difference between the Lufthansa account tweeting out a statement, and a Lufthansa rep, named in the tweet, replying to one user. The reply means that it's hidden from anyone who isn't either following both users (unlikely) or actively seeking it out.
I see this interaction as closer to a customer service phone call where a rep got something wrong, and the only fallout should be that I as one single customer am now misinformed. The amount of checks and balances to ensure that messaging is correct doesn't need too high as it's low risk if one customer is misinformed. Whereas if Lufthansa makes a full statement they should expect all customers to be misinformed, or expect the press to pick it up.
This brings me back to my original point. I don't think the press reported on this well, because this was clearly not an official statement prepared for the press or the audience of all customers, with the level of thinking about it that should go into that sort of thing. It was clearly (to me) an isolated thing. The correct course of action for a journalist would be to confirm with Lufthansa, which appears not to have happened.
I'm not sure what makes you think this is not a customer service interaction?