Arnie says, "I understand that many American pedants insist that, when introducing relative clauses, 'which' must always be used. The British are far less restrictive, however."

OK, dumb question here. What's a relative clause?

Here's how I understand using that and which. Do others differ?

The Bible which is on the table is old. (I also have another Bible; it's not on the table.)
The Bible that is on the table is old. (same)
The Bible, which is on the table, is old. (There's no other Bible.)
The Bible, that is on the table, is old. (We wouldn't say this.)

What's a relative clause?

It's a clause (phrase) that cannot stand on its own as a sentence, (e.g., which is red), and is introduced by a relative pronoun.

If you want to know more about the that/which rule you could do worse than looking through the many entries posted on the Language Log: here.

The Bible which is on the table is old.
The Bible that is on the table is old.

Both are not relative clauses; they are an integral part of the sentence. The Bible on the table is old.

The Bible, which is on the table, is old.
The Bible, that is on the table, is old.

These are relative clauses; they add additional information, (the fact that the Bible is on the table) but the main point of the sentence is that the Bible is old.

All four of your example sentences contain relative clauses. The distinction you are trying to make is one which is usually called restrictive vs non-restrictive relative clauses.