Essentially correct... just a few revisions.
1: The fire that did the damage occurred in the 16th Century (1532AD) not the 14th (1301 - 1400). Previous fire damage (the poker holes) predated the 12th Century since they are depicted on the Hungarian Pray Codex which has a known provenance of 1250AD.2: The area the C-14 sample was cut from was not damaged in the fire; it was damaged by handling, snipping sample relic for gifts to unknown persons in the past, and hanging the Shroud with nails during medieval expositions (even in the 1978 exposition the Shroud was affixed to plywood panels with thumbtacks!).
3: The testing actually tested an amalgam of old original material and newer patch material in varying percentages (depending on the sub-sample's distance from the bottom edge)... which combined to give an invalid dates (1260AD to 1390AD) between the actual ages of both.
With "degrees of confidence" for the four tested sub-samples ranging around +/- 17 years to +/- 30 years, the tests extended the possible invalid dates to 1235 to 1415... which is far too large of an error given the expertise of the labs involved and the supposed homogenity of the sample. In fact, none of the four sub-sample "Degree of Confidence" date ranges, even overlaped the adjacent date ranges. That fact alone should have raised a huge red flag for the investigators; that it didn't shows that the investigators were happy the invalid results "confirmed" their prejudices.
Then they appear to have used the "invisible weaving" technique.
So, damages in the 16th century were repaired with 14th century goods which were then tested by radiocarbon dating and found to be from the 14th century.
There are disputes about how much damage was caused by the fire (which lead, of course, to disputes about which sections are old, and which sections are 14th century), but it's pretty clear that even some sections of the photographic negative were damaged.
I suggest that only those sections with signs of blood, on the body itself, be tested, and then only by whatever sort of non-intrustive technique can ever be developed for that purpose.
Every time "scientific" studies are done on the Shroud, controversy ensues. Such "investigations" accomplish nothing but confirm the doubting in their incredulity and convince the believers that somebody is up to no good.
It seems to me that any tests made will only serve to accelerate the irreparable damage the Shroud suffers anyway due to the passage of time, and in the future this data will be of little use because more sophisticated techniques in the future would have their own methods, independent of current ones. Why inflict damage now? Why not just keep the Shroud in a stable, ideal humidity and temperature, perhaps without oxygen, so as to preserve it as intact as possible for future generations? Isn't that what we would have liked previous generations to have done for our sake?
I visited some museums in Europe about 20 years ago, and the one thing very common at the time was, they prohibited flash photography. They thought that the light exposure would accelerate deterioration of the paintings or other artifacts. I thought it was silly at the time, but in retrospect, perhaps it was really wise. After all, furniture kept indoors but exposed to daily sunshine through a window suffers sun damage over time. So why not things exposed to daily flashes of cameras?
If such prudence is used with things of relatively little significance compared to the Shroud, why not with the Shroud itself? The more poking and prodding it endures, the faster it would perhaps return to dust, and the more accountable we will be in the eyes of our children's children.