I don't know about medieval historians' reception of Livy, but I'm not thrilled the "reliability" of Livy at all. He's a historian of the Roman model, meaning history is a literary activity for him. For instance, a lot of people think his description of the mutiny against Scipio is cribbed from Asinius Polio's description of the mutiny against Caesar. His chronology for dates before 300 (so before the Punic wars, but it still addresses his general reliability) are wrong because of the use of Varro's chronology. He can be bullied by Augustus into endorsing facts he knows are wrong (though he goes out of his way to say EVERY OTHER BLOODY AUTHORITY SAYS AUGUSTUS IS WRONG, BUT HE WOULD NEVER LIE, RIGHT?!?). He's not doing or even trying to do history the way the post-Ranke world does.

He may (MAY) preserve bits of very early information in things like religious formulae and stuff from the priestly annals, but without the annals it's hard to tell how well represented they are.

Now, with all that said, Livy thought he was a hell of a lot different from Homer:

Events before Rome was born or thought of have come to us in old tales with more of the charm of poetry than of a sound historical record, and such traditions I propose neither to affirm nor refute. There is no reason, I feel, to object when antiquity draws no hard line between the human and the supernatural: it adds dignity to the past, and, if any nation deserves the privilege of claiming a divine ancestry, that nation is our own....

He knows well enough that the early traditions involving the gods etc. are not accurate, but they make nice stories so he keeps them.

They certainly would have been aware of the differences between straight history and historical fiction, but didn't necessarily have anything else to work with. English writers in particular were keen to be seen as part of a tradition of 'reliable' history dating back to Bede which meant a great effort to cite their sources. The only trouble with this is that they often go on to make things up anyway. A prime example is Gerald of Wales' History and Topography of Ireland which he opens by stating that everything he writes he either saw first-hand or got directly from somebody who did, and then proceeds to tell some entirely outlandish tales.

Most writers would have had a florilegium which would have contained important quotes from major classical writers and historians for them to use as a reference work and insert when necessary. Gerald of Wales was keen on quoting Ovid. Unfortunately, much to the confusion of the 1978 translator and editor of his Itinerarium Cambriae and Descriptio Cambriae, many of those references appear to be entirely made up (Lewis Thorpe, 1978).

Adam of Bremen's History of the Archbishops of Hamburg-Bremen consciously copies Pliny the Elder's fantastical descriptions of wild Africa from the Historia Naturalis to describe what lies beyond the frontier of Christendom in Sweden and Russia, as a fancy way of saying "I have NO idea what's out there." Which is weird because about a century earlier, Thietmar of Merseberg writes a detailed and seemingly quite accurate description of some of the same area which would have been well known, and Norway and Denmark at least would have already been literate Christian kingdoms.