"Fireproof netting"??? I've never heard of such a thing. If it's netting, then there are "holes" in it AND THEN ITS NOT FIREPROOF. The netting might be fire retardant, but it's not a fireproof membrane. We worry about the exposed surface to make the roofing "fire retardant".
It might be an "isolation fabric". They are used between the roofing and the old roofing. (You want the new roof to move independent of the structure.)
The second roofer's "peel and stick" type roofing has problems too: 1) most are not UV protected (I'd be skeptical and demand to see the "Product Data Sheets".) They are made to be installed under another type of roofing, (I.e.: composition shingles, metal roofing, etc.) 2) When installed over an entire roof, it doesn't allow the roof to breathe. Usually, this type of roofing is used in valleys, at eaves for ice dams, around difficult areas like vents, etc. in low slope areas, etc. and then covered with another type roofing. 3) When fully adhered, it will move with the roof structure, which can put stress on the membrane and it could tear (fail) over time. The second roofers statement about its fire suppressant because it doesn't use nails is wacky.
In low-slope applications, I like thermoplastic (TPO) or PVC roofing...never use EPDM roofing...it can come apart over time. The first roof sounds like one of these (TPO or PVC) with a "mechanically fastened" installation, (nailed down).
If fireproofing is important, I'd use metal roofing with a low-slope application, otherwise I'd use TPO or PVC roofing with mechanical fasteners and "heat-welded" to the fasteners for low-slope applications.
(Btw, if you live in a high wind area, make sure you get the proper number of fasteners per square foot. Check with the manufacturer.)