Most commercial pint- and quart-size mayonnaise or salad dressing jars may be used with new two-piece lids for canning acid foods. However, you should expect more seal failures and jar breakage. These jars have a narrower sealing surface and are tempered less than Mason jars, and may be weakened by repeated contact with metal spoons or knives used in dispensing mayonnaise or salad dressing. Seemingly insignificant scratches in glass may cause cracking and breakage while processing jars in a canner. Mayonnaise-type jars are not recommended for use with foods to be processed in a pressure canner because of excessive jar breakage. Other commercial jars with mouths that cannot be sealed with two-piece canning lids are not recommended for use in canning any food at home.
That is from the National Center for Home Food Preservation which is basically a clearing house for USDA rules. They have a lot of good, tested, authoritative info and are well worth bookmarking.
As a practical guide, you can use most any jar that the two piece canning lids will fit correctly. The rubber sealant and canning lids, both one piece and two are no longer made with a film that is at all reliable for more than one usage. Attempting to reuse them, especially the commercial one piece shown in the question is not safe and will lead to high numbers of seal failure. But if the jar is the correct size for a 2-piece, even, and undamaged, it can be used.
Know though that these recycled jars are not the same as years ago. The thickness and quality of the glass is less to save the packers money and weight and since most will be trashed anyway. They are far more prone to breakage even with tiny scratches you may not even notice. They even can be used in pressure canning, but only if you are prepared to lose some and you will get breakage and failures. It will happen with hot water bath too, just not as often. Use caution though, because sometimes the sizes are not exactly the same as mason type and the slight difference will cause a higher seal failure rate.