According to the BBC article, Self-healing rubber bounces back, "...break a rubber (or most other solids), and the chemical welds - known as covalent bonds - are also broken"
I've never heard from chemical bonds being called welds before, and with good reason! A weld is a join formed by melting two pieces of material together. Having spent a lot of effort creating an atomic structure to make your material strong (by using just the right material and producing it in just the right way), melting it by welding mixes up the atoms, so the layout of atoms is lost and the weld is usually weaker than the rest.
In contrast, a covalent bond is simply a force holding atoms together (arising because the electrons are happier if the atoms are together). If you have two atoms held together by a covalent bond, the bond has a particular strength - The quality of a weld varies depending on how much of the material you melt and how the material reacts to being heated up (and cooled down again) REALLY fast. But a covalent bond has a particular strength irrespective of how you make it.
So covalent bonds are not a weak link, not caused by melting and not variable in quality. A weld is just not a good analogy. If they don't want to call it a chemical bond, call it a chemical link or just a force binding the molecular chains, surely people understand that just as well as "weld". After all, what fraction of brits have tried welding for themselves? Well, on the bright side, they didn't call it a glue!
The material they have made sounds pretty amazing, I really want to look up the paper and read it properly, because a hydrogen-bonded rubber, diffusing quickly back into itself to regain its strength is a pretty cool invention