Yes, public sector refers to government run entities like the post office, also anything staffed by civil servants, and to a far lesser extent businesses contracted to the state.

Publicly traded businesses do have a lot of reporting responsibilities, also the CEO is essentially obligated to pursue profits over all else on behalf of the shareholders. A privately traded business does not have these obligations. Musk made this change with the purchase of the company (but again, he did not "buy all the shares", he bought most of them, $5bn was paid by other parties compared to his ~$26bn [plus a couple bn in fees]) however Twitter as a business was always and still is private sector. This means they absolutely could have censored Trump any time they liked - or anyone else for that matter - and they still can. It's just Musk has skewed the business to one political side, and now Musk doesn't really have to answer to anyone. Even the lawsuits against him and Twitter he'll likely be able to weasle out, because it's a limited company - although I hope they do manage to make it stick, he personally made promises that the purchase was conditional upon, which he has since broken.

So, like I've said from the beginning, it has always been private sector. However, even as a private sector business, it serves as a public forum - it was always a "private business cum public forum" - but now Musk has ruined the public forum part by making it very apparently biased towards right wing extemism.

A better analogy of what Twitter was is a public house. A private sector business, but open to the public (although Twitter never had the licencing regulations that pubs have). Musk has taken over the pub and is running it into the ground, driving out the peaceful regulars in favour of unsavoury people that spill out on to the residential streets and vomit everywhere.

I never actually wanted to talk about publicly traded vs privately traded, you brought that up.