How to choose verb after "there" in beginning of sentence?

Cambridge "Advanced Grammar in Use" provides following rule in Unit 95C:

If the noun phrase consists of two or more nouns in a list, we use a singular verb if the first noun is singular or uncountable, and a plural verb if the first noun is plural:

• When I opened the fridge there was only a bottle of milk, some eggs, and butter.
• When I opened the fridge there were only some eggs, a bottle of milk, and butter.

But Grammar Girl in episode 278 Oddness When You Start a Sentence with "There Is" gives completely different explanation:

A listener reader named Joe wants to know whether he should say, "There is a couch and a coffee table in the room," or "There are a couch and a coffee table in the room."

...

It's a compound subject since it has two nouns connected by the word "and," which makes it plural ... Now that you know the subject is "a couch and a coffee table" and that it's plural, it's easy to choose the right verb: "are."

I'm somewhat confused by these contradictory rules. Could someone explain what I'm missing here?

This is a very good question. I think the difference is between a comma-separated list of three or more subjects, and a simple ("and"-separated) two-part compound subject.

Let's take the slightly longer list: "When I opened the fridge there was only a bottle of milk, some eggs, a loaf of bread, and butter." In such a list, the items form clauses in the sentence; each one could be the singular subject or object of the exact same sentence: "... there was only a bottle of milk", "...there was only a loaf of bread", etc. You could also remove either of the two inner items with no change to the rest of the sentence (the fact that I was able to add the extra item to your quoted sentence without changing anything else in that sentence demonstrates this). In this case, the first rule you stated is correct; you pick the verb conjugation that works for the first item in the list, as if it were the only one.

However, when you get down to two items, now there are no more commas. The two items, say "a bottle of milk and butter", now form a compound subject; the two items are being referred to as one entity, which is always plural. In this case, the plural verb should be used.

EDIT: Good points. Some of those sentences sound better than others:

There is further rain and strong winds forecast for the next three days. - Not bad. I think "are" works here too.

There was a loud bang and some flashes of light before flames started pouring from the windows of the house. - This does indeed work better with "was" than "were", no question

There is a bank and cash machines in the city centre. - this one grates my ears; "there are a bank and cash machines" sounds much better for some reason.

There was no water or animals anywhere in the desert. - OK, but I think "were" sounds better here as well.

The biggest question, given the above, is why there is a difference between the second and third sentences; their structure is practically identical, the only difference I can detect is tense.

It depends how you parse the sentence. Taking Grammar Girl's example first, you could read it as...

There is a couch and [there is also] a coffee table in the room OR

There are [two items, namely] a couch and a coffee table, in the room

Neither interpretation is any more 'correct' than the other, since semantically they convey exactly the same meaning. Therefore both is and are can be validly used (and in fact are, though many speakers will decisively and vehemently argue for their own particular usage).

The Cambridge Grammar deals with the very specific context of a list - specifically one wherein the elements are baldly set down with no interconnecting and's. In that context there is no question but that the verb should agree with the first noun in the list.

LATER: I should just add that in the case of a couch and a coffee table I imagine most speakers would favour the singular verb. Some of those might change their minds with, for example...

"Where's my fork? There are just a knife and spoon here."