Having given you the background over valuation of private equity, following are the answers to your questions.

Why would investors "mark down" their own stakes?

Funds have invested in the company for a specific time horizon after which they would need to sell their stakes, exit and distribute the money they get to their own investors.

So, at any point in time, the valuation that they make of their holdings should be in sync with what they can actually get on selling their stakes. If they artificially inflate the value of their holdings, when they ultimately sell the stakes, they'd receive a far lower amount than what they had valued their stakes at and that would anger the investors. So, if the real market value of a company goes down, the funds need to show that in their valuations as well.

How are these "mutual fund investors" different from investors who give these ecommerce companies money?

Some mutual funds do have a limited mandate to make illiquid investments like real estate of private equity up to a maximum proportion of their NAV. These funds like many other hedge funds make late stage investments in "start-up" companies which are usually 2-3 years away from IPO which can be a big pay day for them.

A big difference between these mutual funds and normal venture capital would be in their disclosure norms. VC funds don't have to make public announcements about the value of their holdings, however, mutual funds have to. Hence, they have to release the periodic valuation in their holdings in stocks of privately held companies. Which is why you hear about mark up or mark down of the valuations of these companies even when they are not raising funds.