Does the HaplotypeCaller method rely less on other samples to make calls (in the GenotypeGVCFs step) than the UnifiedGenotyper?
Haplotype Caller actually relies more on other samples' data to make calls than Unified Genotyper. Please read more about the joint genotyping here: https://www.broadinstitute.org/gatk/guide/article?id=4017
Is there a way to run the GenotypeGVCFs tool in a way that makes no attempt to correct any calls coming from the HaplotypeCaller - basically to run it as a gVCF->VCF converter - for comparison's sake?
To elaborate slightly on Sheila's answer, the "reliance on other samples' data" (which we prefer to view as an empowerment rather than a dependence: you get good calls alone, but even better calls together) is essentially the same behavior between the HaplotypeCaller ref confidence/GVCF method and UnifiedGenotyper run in multisample mode. But the maximum potential extent of the empowerment is greater in the HC method because it enables you to use much larger cohorts. So it's not that either is more or less dependent on cohorts, but that HC can give you a bigger bang if you have a bigger cohort. (On top of making better indel calls whether you use a cohort or not, thanks to the reassembly step.)
You can certainly run GenotypeGVCFs on a single-sample gVCF, in which case it will have no cohort information to use. That will mimic basic single-sample calling behavior. You may see marginal differences if you compare that to just HC run in basic mode on a single sample, but nothing that would make a real difference between the two callsets after filtering. If you try this, do let us know how it behaves in your hands of course.