According to your second link, the Rajamangala Institute of Technology made a 16 layer silk bullet proof vest that could stop a 9 mm round.
A Class II vest intended to stop most 9mm rounds is made of ~21 layers of Kevlar 29 on average, making it ~12.6 mm thick. I'm completely failing to find a thickness for a grade of silk, but assuming it's equivalent to Kevlar 29, it would be 9.6mm thick, so based on that assumption and if spider silk is equivalent to 'normal' silk, I'd agree that it's more comfortable.
However a Class II vest isn't rated for knives thus a 9.6mm thick silk vest still wouldn't be either; a 1.5 - 2mm thick rigid steel cuirass will definitely stop a knife, so I still find your original statement that a spider silk vest would be comparable to steel armour is dubious at best. Steel armour and Kevlar are intended to stop different threats, which is the hyperbole that I'm objecting to.