Well, the way I look at it, VC is just a form of private equity. Let's start from the basics of private equity. Essentially, you raise private money to invest in something. Because it is equity, you get certain tax benefits (maybe not for long of Obama is elected) and because it is private you avoid certain filing and reporting requirements. Without going into too much detail, in order to sell securities they need to be registered, sold through broker-dealers (who are fiduciary requirements) and must be accompanied with a prospectus, and a whole bunch of other stuff. There's a limit on who can buy certain securities, and how they can be marketed. With private equity, investors with a certain level or wealth are assumed to be informed investors, and disclosure requirements are relaxed or waived. A key feature of private equity investment is that there is usually a lock-up period, perhaps 5 years where investors cannot get their money back, and after that period they can withdraw their share of the investment at certain intervals (every 6 months or something) by giving written notice in advance. This allows a private equity fund to invest in different, illiquid stuff, making it different from a mutual fund or something like that.
In recent years, private equity has come to equal leveraged buy-outs (LBO), but really, a private equity fund can invest in anything it wants to. It might market a certain strategy to its investors which will guide its investments. LBOs were all the rage for several years.
But a hedge fund is essentially private equity as well. It's money from private investors, held in a fund with a lock-up period, and invested in almost anything, from public equities to real estate to hard assets to whatever managers can think of.
And venture capital is pretty much just a type of private equity fund that focuses on start-up companies. They might take a more hands-on approach, provide connections, give advice, and things like that, but at the root they are basically establishing investment funds with private money and investing using a certain strategy. For tech VCs, an engineering background is probably very nice because it allows you to understand their business, but also because, quite frankly, engineers are more likely to listen when approached by one of their geeky brethren (just kidding to all you engineers that just blew your tops). It's the same reason why a fund investing in the defense industry might want a former politician on their staff.
Certainly, there are differences in the three types of funds, but it's not unusual for a HF or late stage VC fund to be set up very similarly to PE funds.
So, getting back to your original question, the three types of firms are similar in many ways; but I agree with you that people interested in PE/HF fall back to banking, while people interested in VC fall back to entrepreneurship or some other role. I believe the reason why they are grouped together is that people view they as very similar in terms of the potential for outsize pay, risky and difficult recruiting, as well as other things related to job hunting. Tech might be what comes to mind when we think of VC, but really, there is venture capital in every industry. If I'm running a VC fund focusing on entertainment companies, am I really going to favor geeky-fresh engineers (don't hate me

)?