The website has lots of information and details, describing both....
Please check about bad sectors:
(2) https://www.hdsentinel.com/help/en/57_desc.html (or click on the ? button in Hard Disk Sentinel, next to the text description) and check the "Bad sectors" section
About weak sectors:
Without repeating and quoting, I try to describe:
The BAD SECTORS are no longer used by the hard disk drive (or SSD). Instead, the disk drive uses a special spare area for all reads/writes, so if we try to read/write the BAD sector, then we actually read/write this spare sector.
As a result, BAD SECTOR is no longer used and no longer causes problems. After making sure that all such bad sectors are fixed this way (which is called reallocation), then we can even acknowledge the count: to prevent these bad sectors displaying (as they are already fixed). So then Hard Disk Sentinel will only report possible new issues (if there will be). This is described on (3)
In contrast, WEAK SECTORS are still used by the disk drive. Reading results errors in surface testing, chkdsk INCORRECTLY shows these are bad sectors (and attempts to hide them instead of fixing) and internal self tests fail because of them (see (4) for all details).
These can be usually repaired by the
Disk menu -> Surface test -> Disk Repair (if the count of problems low, keeping all data)
Disk menu -> Surface test -> Reinitialise Disk Surface (which performs complete erase, like "low-level-format") This forces the hard disk perform reallocation if required (which may cause bad sectors if required, as then the disk will no longer use them) and repairing weak sectors too.
> (Q1) What's the difference between a damaged sector and a bad sector?
See above. Damaged sector = weak sector = pending sector (all but different name) while Bad = sector which is no longer used.
> (Q2) After a repair or re-initialization test, there should be no weak sectors because they
> have all been reclassified as good/damaged/bad?
Ideally yes, if all weak sectors found and repaired then their count should be zero after the test.
> (That's why tests don't state the number of weak sectors?
This is not true....
You can check the possible remaining weak sectors on the Overview page or the S.M.A.R.T. page in Hard Disk Sentinel.
Also after the test, the summary window shows how many weak sectors fixed/stabilized.
> (Q3) It appears impossible to know where the weak/pending sectors are. Correct?
Hard disks do not provide their location (which is sad) this is why we'd need to perform a surface scan to reveal and fix them.
> (Q4) Can't do a quick test that only tests the weak/pending sectors? (or does that happen automatically in the short tests?)
No, it is not possible. The short test usually fails quickly (see above link) even if one weak sector found, regardless of its location.
> (Q5) Bad sector count will only appear once all spare sectors are used (or otherwise can't be reallocated)?
> (Sectors can be reallocated anywhere or only within the same track?)
Sorry, I do not understand what you mean.
There is a spare area, eg. 1500 sectors (just an example) which is not used otherwise by the disk drive. When a sector needs reallocation (for example the Reinitialise Disk Surface test forced the hard disk to do so) then the original sector is no longer used and all further reads/writes are redirected to this spare sector. Then the bad sector counter on the S.M.A.R.T. page increases, showing that a bad sector appear.
Until the spare area is filled, we should see no problems by these bad sectors (and no space decrease or similar).
The location of the spare sector is defined by the manufacturer, it may be at completely different track.
You may ask the manufacturer of the drive for details of that particular model.