I'd firstly caution that "I think he is the man who/that did it" does not show an
essential use of a defining relative pronoun (rather like the "Who is the sumotori who is Yokozuna?" example in the "The worst relative pronoun exercises you've ever seen..." thread!); I imagine it would be reduced simply to, "I think HE did it!".
Similarly, "He is the man (that) I like" is a little strange (it is a marked word order for the alternative "I like HIM"...or, at the risk of blurring the line between demonstratives and relative pronouns, perhaps even "I like THAT man"

).
Let's instead look at the title to a James Bond movie:
The Spy who loved me. Here we definitely have a "subject" loving the object "me", and if we remove the "who" we end up with a simple, story-like statement about apparently only one spy (as opposed to from a potentially infinite number of spies in the "who" sentence). This is all clearly very different to "The spy (not the doctor!) (who/that) I loved...".
I have found that a very useful way to make students more or less HAVE TO use relative pronouns is in talking about actors:
A: Do you know David Warner?
B: No, what's he been in?
[A: He was the bad guy in
Titanic.
B: You mean, he was the guy (that) Rose was going to marry/?who was going to marry Rose?]
A: He was the bad guy (in
Titanic) who put the necklace into jack's pocket, and who later handcuffed Jack to the pipe!
Now, obviously David Warner has played more unique roles: if students were looking at a photo of him they might also realize that:
He was the theatre director in
Scream 2.
He was the evil wizard in
Time Bandits
etc.
The point is, that where there is (conceivably) more than one person with that role in the movie, the defining relative pronoun becomes useful, and we define the person according to the things they did (or that were done to them), in order to distinguish them from the group they are a part of.
Basically, we want an environment in which we (I or you), and the people we know, cannot enter and are thus not the main actors, not the main movers and shakers. Only then can the "other people" really become the grammatical subjects themselves! (See how the "bigger" role of Rose could well push the "bad guy" out of subject position, it is not just because women (can) choose to not accept a marriage proposal, but because she is the more major character, I think. "Construal"
* is a very interesting area!).
Another example: Do you know Tom Sizemore? He was the astronaut who blew himself up in
Red Planet. He was also the bank robber who grabbed the little girl (and who got shot by Pacino) in
Heat. Oh, he was also the sergeant (I think) in
Saving Private Ryan.
* "Construal: The process whereby a given state of affairs is structured by a language user for purposes of its linguistic expression. Typically, a state of affairs can be construed in alternate ways." From the Glossary in John R. Taylor's
Cognitive Grammar (OUP 2002).