Both of those methods are accessing the public markets. Private investment means investing in non publicly traded companies. Like venture capital... For example

That kind of investment requires having a large amount of capital to begin with.

There are many websites that cater to private investments. Usually you need to be an accredited investor. But you don't need huge amounts of capital.

https://angel.co/ for example.

Or being a founder / early employee of a company that ends up being successful. You'd invest your time / energy / knowledge in that case.

You can invest in listed PE / VC funds and lots of larger actively managed funds have some PE element.

For example in the UK Electra and III are PE funds and Witan in the UK as 10 or 15 % in private investments.

And that's not including UK specific things like VCT's where you would need to have a few k to invest