Not specific enough in terms of jobs I'm looking for?

Hi guys,

I've worked 3 years in large cap PE fund (I'm associate now) and I'm considering a move (mainly for relocation purposes).

If I'm being honest, here's what I'm looking for:
- Don't care about the industry (except infra/RE/FIG)
- Don't care about investment size / stage: VC / Growth / Large cap
- Don't care about where the money comes from (PE fund, sovereign fund, pension fund, ...)
- Care about investor type: I want a lead investor, not a passive co-invest or fund of fund
- I want their approach to be as hands on as possible in terms of management

I've mentioned this to a few headhunters who said I'm not specific enough. Fair enough for investment size, being associate at a VC firm is a very different job form associate in a large cap fund, and requires different skills. But for the rest? Is it really an issue and should I lie to every people I meet based on the opportunity they offer?

To me, fit with the team is waaaay more important than anything else, including salary, working on a specific industry, or knowing that the money I invest comes from LPs rather than a state fund. Or should I just ignore those guys because that's definitely not how people think.

Any advice on how to express that better is welcomed :)

Thanks!