Venture capital is a form of private equity.

We are, in fact, conversing on a website owned by a private equity firm.

VC can extract value by selling it on at an inflated valuation.

That's what SoftBank tried to do with WeWork. The company is worth 10 billion (or less) and they tried to sell it to the public markets at 40 billion. Had the public markets not noticed the weird shit going on inside the company and actually bought it at 40+ billion then SoftBank would've made a lot of money. The stock would then most likely decrease over the following years costing those that bought it in the IPO money.

This is why it's so important that public markets are strictly regulated and the SEC forces companies to publish all kinds of things before IPO.

Private equity and VC are generally interchangeable terms in that context. There's a difference between a VC that privately invites certain investors to invest, versus a publicly open mutual fund that anyone can buy into.

And to complicate matters even more, there are publicly traded private equity partnerships (Blackrock, etc).

tldr, jargon sucks