Private is used in opposition to public, as in 'private sector' and 'public sector', where public sector is ran and funded by the government, which means public finances, which mainly means taxes - people cant pick and choose which taxes they pay and what the money is put towards. In private sector on the other hand, people can privately decide what to invest their money in.

Sure, when you put subsidies and tax breaks into the mix the line gets a bit more blurry, but legally speaking its a well defined concept, even if not all private companies match the intuitive meaning of the phrase. And yeah, Google is a very specific case, which is why there is always some level of debate about whether it (and other similar companies) should be forced to split up, and some of the resulting parts turned into a public utility service.