Alright so I'll breifly explain this and the difference real quick to you.
State Corp- IS a private corp, but it is owned by the state. 100% of ownership is the country and the country makes the decisions. And is entitled to all the profit, but you can be creative how you prefer to collect it.
National Corp- Same as state but it is MORE private because it isn't possible to sell shares.
Country Controlled Corporation- These are public, you lose the ability to completely freely manage the money in or out, but this happens when the state owns the largest active amount of shares. The largest amount that is not owned by an Investment Fund, or foreign country. You do not receive a efficiency or quality max upgrade, unless your country is the largest shareholder and is under 25% ownership.
CEO Controlled Corporation- These are public and very similar to Country controlled corporations. There are 3 big differences the first is that the corp now has to pay out Country Resource Payments. This uncontrollable fee limits profits in respect to the corporation, not necessarily the country however. The corp will lose the profit to CRU and should have a lower MV. So if you want to be nice to your CEO let the country have it, if you want to be nicer to the country let the CEO have it. The second difference has the ability to trump the last statement, they have a maximum efficiency and quality boosted by 25. Allowing you to sell for higher prices, and for a smaller work force. The third difference is... did I say 3 i meant 2... maybe a just had a small brain fart. Or i forgot the third oh well. The controller is determined the same as CCC.
Finally we have the Privately Controlled CEO Corporation. I think this is what you are looking for. It is kind of a hybrid of state and CEO controlled public corp. As it has to pay CRU has a max Q&E of 225 but can't be increased to 250 like Public Controlled Corps. You gain back the ability to fiddle with the money. The way to make a corp Private like this is to collect 100% ALL of the shares. If people are able to buy shares how is that private?
So I think that answers your question, if you want it to be private collect ALL the shares for your CEO, but state corps are also private, giving your situation that probably isn't what you are thinking though. But Liberation Peace, it is very important to understand why you want to do what you are doing. Typically people try to get these corps public for various reasons, it rarely goes the other way. The process of turning a state corp into a private corp through an IPO is probably an annoying process. In the future you may want to just build a corp with your CEO, failing that, buying the corp outright when it is still a state would be a far arduous process. Anyways good luck to ya.