I suggest first that historically, "Apache" when used alone refers to the web server, and indeed still does. Consider the release announcement for 2.4.9 currently on the project's home page at http://httpd.apache.org/: (Other release announcements on the page call it "httpd" for short.)
The Apache Software Foundation and the Apache HTTP Server Project are pleased to announce the release of version 2.4.9 of the Apache HTTP Server ("Apache"). This version of Apache is our latest GA release of the new generation 2.4.x branch of Apache HTTPD and represents fifteen years of innovation by the project, and is recommended over all previous releases. This version of Apache is principally a security and bug fix release.
This usage is probably because Apache (the web server) has been the most popular web server on the Internet since before the Apache Software Foundation existed.
I think that means it'd be hard to make apache mean anything but the web server; it'd be a constant battle against new mistagged questions.
Also, most of the questions people would probably have about the Apache Software Foundation (how does my project join, how do I join an ASF project, etc.) are off-topic here. The relationship between Apache projects is non-technical as well. Just because you're an expert with Ant, doesn't mean you have a clue about OpenOffice. It's no more a useful tag than, say, "microsoft" would be. If there are on-topic questions about the ASF, then it seems apache-foundation or even apache-software-foundation would be fine.
So, in summary:
- Tomcat should probably be apache-tomcat, probably with a synonym from tomcat. Or maybe the other way around; only the latter actually has questions.
- Ant should probably be apache-ant. There aren't any questions tagged ant, so I see no need for a synonym there.
- If we do decide to retag the web server to apache-httpd then we need to get rid of apache entirely. Otherwise, we'll have to be constantly retagging. Wouldn't be surprised if we even need it blacklisted.
- We can use apache-software-foundation or apache-foundation if we find any on-topic questions actually about the ASF. (I have no idea if the ASF would accept a project named "Foundation". If they did, then the latter tag would be confused again.)
- I don't see any reason to tag a question ASF just because it talks about an ASF project, any more than I think it'd make since to tag one Microsoft (more on Stack Overflow or Super User, of course).