When does the TLD having glue records for the nameservers save DNS lookups?

To add to Mit's answer. The best practice is to only use Glue Records to resolve those circular dependencies. See section 2.3 of RFC1912: http://www.faqs.org/rfcs/rfc1912.html

To quote:

Some people get in the bad habit of putting in a glue record whenever
they add an NS record "just to make sure". Having duplicate glue
records in your zone files just makes it harder when a nameserver moves to a new IP address, or is removed. You'll spend hours trying to figure out why random people still see the old IP address for some host, because someone forgot to change or remove a glue record in some other file. Newer BIND versions will ignore these extra glue records in local zone files.

Note that last bit if you're using BIND. Also, I think most DNS clients will ignore extra-domain glue records, which would explain what you're seeing.

Edit to follow up on comment.

With TLD glue records, you'd typically provide them to your registrar, but otherwise the same rules apply. I run domains in .co.uk and .com. I also have name servers in both of those TLDs that are authoritative for all of my domains.

If I run dig ns co.uk and dig ns com, I can see the big heap of servers that are authoritative for those TLDs. If I pick one of those listed .com servers and dig @<server> ns <mydomain>.com it'll return all four name servers for my domain, but only the glue records of the name servers in the same TLD (supplied as ADDITIONALs), which is what you're describing.

If I run dig @<co.uk server> <myotherdomain>.co.uk, I'll again get all four name servers, but this time accompanied by the glue records for the 3 of them in the .co.uk TLD.

This is all as it should be. If all of your name servers are in the .us TLD, then clients looking for your .com domains will be told the domain names of your name servers by the .com servers; then the clients will perform another query on the .us servers to find their IPs. That extra round trip might sound inefficient, but it only has to happen once per TTL period per client. (typically 48 hours for TLD records.)

(Apologies if you already know all of the above). To answer the original question. Glue records are not intended to save DNS lookups. They're necessary to prevent infinite loops.

Edit 2

Sorry for waffling around the issue. I see what you mean, now (Good diagrams, by the way).

My understanding of how clients should behave is that if one has the opportunity to get authoritative RRs, then it should; but in this case, the client is asking the name server for its own A records, which is pointless, since the Glue is those A records.

I'm reaching the limits of my knowledge here, but I can't think of any situation why a client would need to do this. I'm guessing here, but perhaps the client implementation needs to check that the server is definitely authoritative for ns_.host.us, even when that means what looks like a pointless, extra round-trip.

Try a dig +trace www.example.com to see if it does the same thing. I'd be curious to see if it does the same lookup if the original query was for a _.host.us name, too.

Name servers in delegations are identified by name, rather than by IP address. This means that a resolving name server must issue another DNS request to find out the IP address of the server to which it has been referred. If the name given in the delegation is a subdomain of the domain for which the delegation is being provided, there is a circular dependency. In this case the nameserver providing the delegation must also provide one or more IP addresses for the authoritative nameserver mentioned in the delegation. This information is called glue. The delegating name server provides this glue in the form of records in the additional section of the DNS response, and provides the delegation in the answer section of the response.

For example, if the authoritative name server for example.org is ns1.example.org, a computer trying to resolve www.example.org first resolves ns1.example.org. Since ns1 is contained in example.org, this requires resolving example.org first, which presents a circular dependency. To break the dependency, the nameserver for the org top level domain includes glue along with the delegation for example.org. The glue records are address records that provide IP addresses for ns1.example.org. The resolver uses one or more of these IP addresses to query one of domain's authoritative servers, which allows it to complete the DNS query. This might fix the issue.

Glue is intended to bootstrap queries which would otherwise be unresolvable. Imagine no glue:

1. query root for www.example1.com, get referral to com namesevers.
2. query com nameserver for www.example1.com, get referral to ns1.example.com, the nameserver for example1.com.
3. Oops, we don't have the IP for ns1.example1.com. query com nameserver for ns1.example.com.
4. goto 3.

The com nameserver needs to return an A record for ns1.example.com or you never get there.

You can still run into problems with out of bailiwick delegations since authoritative nameservers don't often return glue records for those. For example:

1. query com nameserver for ns1.example1.com, get answer of ns1.example1.us
2. query us nameserver for ns1.example1.us, get answer of ns1.example.com
3. goto 1

Whoops -- neither the com nameserver nor the us nameserver will return glue for out of bailiwick referrals. DJB has some examples in more detail.

Anyway, glue is not intended as a shortcut, but as a mechanism to avoid loops. Caching provides your mechanism to save lookups.