HD Tune is for HDD not programmed for SSD, for SSD to have bad sectors is normal, means that 1 sector is considered as worn out and replaced by spare sector.
SSD does carry more sectors than normal HDD would, normal HDD only carries around 1024 spare sectors so for HDD to have bad sectors could means physical damage, also means it's faulty.

Despite having a limited amount of spare sectors for the HDD, having a few bad sectors on the HDD can also be normal. It is a cause for concern if the bad sectors increase quickly.
Both are actually reporting the same thing, CrystalDiskInfo reported the same value '1' in the Raw Values. And I also think that both are correct. HD Tune is correct in that this is something to be concerned about, but CrystalDiskInfo is also correct that your SSD is still good at the moment. The ability to reallocate sectors is a self-healing feature intended to cover the occasional minor bad sectors that do not require manufacturer intervention.

For now, just watch the values. If nothing changes over the next few months, then the automatic "healing" has merely done its job and it is not a problem; I have a few HDDs like this and they are still good years after. On other hand, if you have a zero tolerance policy and you don't mind not having the SSD/HDD for the next few days or weeks and losing the data on it (you are likely to be returned with another SSD/HDD, check the serial numbers), you may be able to send it in for warranty without too much trouble. Of course, the technicians may roll their eyes at seeing yet another SSD/HDD like this, or perhaps they are already too jaded at seeing thousands of perfectly fine SSDs/HDDs with one or two reallocated sectors to even bother.

Edit: There is also your warranty period to consider. If it is going to expire soon, it might be a good time to send it in. If you are lucky and they no longer have stock of your model, you might get back a 120GB SSD.