This is not the first time I see this exact pedantry on HN and I can't help but feel the same way as with people who immediately yell "it's not free! Nothing in life is free!" when you talk about free healthcare or free public transportation.

We know. We understand that it is technically correct. But it's completely useless in the discussion. You know perfectly well what the other person meant.

You are technically correct - but it's overly pedantic. Another commonly accepted definition of public companies outside of the US is companies that are partially or wholly owned by the government and thus act as a public service. This includes crown corporations in Canada or the USPS in the US. I think within the context they used that calling Microsoft a private company is fair since they have no obligation to "act in the public good".