Can "that" introduce a clause that contains unecessary information?

Non-restrictive relative clauses -- the kind that take comma intonation and use which -- are not grammatical with that. If you omit the comma intonation and use that, it's no longer a non-restrictive relative clause, but rather a restrictive relative clause, which has different syntax, and the different function of defining the noun it modifies, instead of giving extra information about its antecedent.

In the case of the two example sentences, (1) has several senses that (2) lacks. For instance, in (2) the restrictive relative clause with that must modify the noun phrase a lake.

However, in (1) the non-restrictive relative clause with that could refer to the noun phrase a lake, or to the whole clause a priest drowning the girl in a lake, or even to the resultant proposition that the girl drowned in a lake. These senses are not available with (2), but they change the interpretation of the rest of (1).