Note: even though the phrasing might not make it clear at every turn, this applies to any and all "diversity markers", for lack of a better name; whether it be race, gender, sexual orientation, disability, neurodiversity, ... Repeatedly listing the whole lot became obtrusive in terms of readability, so I mainly used racial identity as the default example, but the core of my answer applies to all "diversity markers", not just race.
You're asking for a universally accepted answer on one of the biggest political hotbeds of the 20th/21st century. The short answer is that there is no universally accepted answer here.
This isn't a "racist vs non-racist" bipartisan issue. Even if you only observe opinions that are considered "non-racist", opinions still wildly differ on where the line should be drawn or what is considered freedom vs what is considered abuse.
In the absence of such a universally accepted approach, your company has to make a provisional choice on how to approach this topic, based on how they want to operate and be publically perceived.
I'm not saying your company should roll over and let the diversity system be blatantly abused. I'm also not saying your company should enforce its own ideas on racial identity and how it reflects on their pro-diversity culture.
What I'm trying to point out here is that this is a really fine line to balance which requires personal considerations, and due to it being heavily politicized it's nigh impossible to not offend some people along the way, regardless of whether their offense is based on fact or public perception of your company.
- How can recruiters prevent candidates from faking diversity, when it appears to be beneficial for the application?
If you assume a context where people self-identify, and where questioning such identification is considered unacceptable, it's essentially impossible to prevent this.
I'm not interested in discussing whether or not the questioning of such self-identification is acceptable or not and how it should be done (if so). There is a spectrum of opinions on this topic. What I'm trying to point out here is that questioning a person's racial identification is going to get a subset of people up in arms, whether on principle or because of a specific case, and if it gains public traction it's going to be a negative PR experience for your company.
No matter your (or my) feelings on the subject, Rachel Dolezal is an interesting facet to consider here. She caused an (inter)national thought exercise about the limits of self-identification (or not) and how to approach it in a multicultural society. That thought exercise hasn't yet yielded a commonly accepted answer that balances the two considerations, as far as I'm aware.
Dolezal's critics stated that she committed cultural appropriation and fraud; Dolezal and her defenders asserted that her self-identification is genuine.
This is essentially the fork in the road for your company. What would your company do if Rachel Dolezal applied for a job?
One thing we can definitely agree on is that you're not interested in your applicants' genetical makeup. Diversity hiring is done for cultural reasons, rather than genetic ones.
But the question is more what you'd rather risk. Do you risk coming across as willing to infringe on a personal freedom (i.e. self-identification), or would you rather risk having the diversity hiring system abused?
And as is the case with all of these kinds of issues, where do you draw the line?
Maybe Rachel Dolezal is considered an edge case and your company chooses to not question her self-identification since it reaches farther than just doing it to game the job application, but at what point do you respond to an applicant who clearly ticked a box without having any remote claim to that being correct and has shown no vested interest in this self-identification other than the personal benefit during the hiring process?
- How can recruiters be inclusive to people who actually fall under their diversity criteria, but do not want to disclose this info to their employer or the public in general?
Much like the core of the first issue, you can't do this without inherently considering yourself (as a company) as a trusted judge of people's racial identity.
Additionally, there are some people from groups who generally stand to "gain" from diversity hires (i.e. minorities or marginalized groups) who still don't like the concept of positive discrimination (I'm intentionally not exploring their reasoning - not my call to make and not relevant here). They will prefer to avoid it and not "play the diversity card", so to speak.
If your company overrides their lack of playing the diversity card, judges them to hold that card anyway, and plays it for them, that is a massive infringement on your applicant's personal freedom.
Your company comes across here as a big checkbox-ticking entity with little regard for employees' personal freedom or self-actualization, on top of being racially judgmental since your company makes up its own mind on what race (or other protected class) it considers specific people to be.