These are called dangling modifiers. It’s the same sort of idea as a dangling particle, but with something other than a participle.
A comma splice has two full sentences joined together with a comma. The first one is not a comma splice, since the bit before the comma has no main verb.
A couple of my favorite dangling modifiers:
“Flying through the trees, I saw a bird.”
This one could be correct, but only if the subject “I” is Superman.
“Running to catch the train, my books fell.”
It would be great to have books capable of running to catch the train.
They kind of look like different things to me: the first one would be fine if it had an additional word, it’s just missing that connector; the second would need a lot more rebuilding.
“With her [warm body] nestled next to him, the bed was warm and comfortable”. Warm body could be taken out (I’m hoping she happens to still be alive), but that’s not a matter of grammar.
Moving the “with” modifier to the front of the phrase emphasizes why the bed was now warm over the fact that it was.
Nestled is a verbal form, but if I’m reading the sentence correctly it’s a participle: an inactive form. If it was the past simple (which for this verb happens to match the participle), then it would be an active form.
For the first, the OP came back to say he intended “nestled” to be an adjective, not a verb. Otherwise, your correction would be correct.
It doesn’t make sense almost in the same way the second doesn’t make sense. The phrase looks like it’s modifying the noun after the comma, so it looks like a type of misplaced/dangling modifier. That’s what the OP is trying to figure out. Now, with the second sentence, the modifier is supposed to go with “sun” (although this is a horribly awkward way of writing it, even incorrectly), but the first doesn’t have anything it’s supposed to be modifying in the sentence. As has been said, it’s missing a word: “With her warm body nestled next to him, the bed was soft and comfortable.”
So really, I’m not entirely sure I’d call the first a dangling modifier, since it doesn’t seem to be modifying anything.
I interpreted “Her warm body nestled next to him, the bed was warm and comfortable” as a preceding nominative absolute. No error. No comma splice. You can see a very famous example of a nominative absolute in the Second Amendment:
"A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed. "
Nestled can also be an adjective, which was what I meant. I’m very sorry to have used a bad example! Totally my fault. (Sentence fragment; good device, will be used again later.)
ETA (I’d never heard of a nomiative absolute before. I’m making errors I never even knew of! Cool!)
In my view the first one is plainly a comma split. There are two independent sentences, and they either need to be written as such, or separated with an appropriate break like a semicolon or, less formally, arguably an em-dash or an appropriate conjunction.
The second example simply makes no sense, though it resembles in form a dangling participle or dangling modifier.
I disagree. A nominative absolute is a standalone phrase that is semantically but not grammatically connected to the rest of the sentence. The first half of this comma splice isn’t a phrase, it’s a complete sentence. Look at the examples in your link. They’re not at all the same.
The Second Amendment and some of the other writings in the Constitution should not be held up as an example of anything, as in those days they sprinkled commas around like decoration. For example, as noted, there’s another comma after “militia” and yet another one after “arms”, and both are redundant and in today’s writing would be considered unnecessary and wrong.
Except, as noted by the OP, they are not, or at least they are not intended to be, given the OP intended the adjectival and not verbal reading of “nestled.” That said, I guess the ambiguity can lead one to read it as a comma splice. From my reading of the sentence as intended by the OP (which is the way I originally read it, too), it’s a textbook nominative absolute.