Do certain registrars have the power to say, exclude an email address, but keep name, location, etc in the WHOIS? If so, I would like to see how differently the top registrars do WHOIS (what information they give.)
Your question is too vague, as this depends on the TLD.
In the gTLD world, the content of whois is restricted per ICANN agreements. You can find all details in Specification 4 at https://newgtlds.icann.org/sites/default/files/agreements/agreement-approved-31jul17-en.html for registries and in section "REGISTRATION DATA DIRECTORY SERVICE (WHOIS) SPECIFICATION" of https://www.icann.org/resources/pages/approved-with-specs-2013-09-17-en for registrars.
Note however that since a few weeks, due to the GDPR, there are a lot of changes going on in whois land and you will see many different cases right now. See this overview: https://www.icann.org/resources/pages/gtld-registration-data-specs-en
Your question may be related to that directly in fact, but you are not providing enough details. If this is the case just have in mind that they are current "power" battles between ICANN, EU jurisdicions, some courts and registrars (like Epag recenly), etc. so that the situation is not clear and a lot of actors made changes that may or may not stick in the future. Also, for the same reason, currently port 43 whois access is more and more shifted towards web access, where you can more easily implement captchas and things like that. Things should be better after this temporary specification under which we are today, if RDAP is indeed introduced as replacement for the current aging whois system.
You also need to understand the registry vs registrar split. Mandatory for thin registries (currently
.NET) where the result, for a given domain name, will be different depending on if you are querying the registry whois server or the registrar one, for the simple reason that in this case the registrar has all contact informations, where the registry has not.
As for ccTLDs, since there is no standard whois format, each registry does its own. You may often found a "Key: Value" pattern, but this is certainly not 100% of cases. And there the registrar whois is not required by registry contracts, but some registrars do have one, and for their ccTLDs domain names they are surely free to implement whatever they wish as format and content.
No, a Registrar couldn't really do that. The Registrar needs to do what the Registry says. And if applicable, the Registry needs to do what a higher authority says (like say ICANN with GTLDs).
At the current point, because of GDPR, most Whois data for most TLDs is obscurred. Prior to that, there were some extensions that might hide Registrant data, (CA, UK are two examples I can think of) but that is/was a Registry decision.