Avoiding foodborne botulism

Food-borne botulism is a risk mainly in homemade products which were not adequately sterilized before extended storage. The reason is that there must be live botulism spores in the product in the first place, which is not the case with store-bought products as long as these are undamaged and were prepared in accordance with hygiene regulations. For example, garlic oil should only be used in store-bought form, as garlic is a known carrier of botulism which thrives under anaerobic conditions. Of course, with the right procedure, garlic oil can also be made by oneself, but it is imperative to adhere to the right methods such as the pressure canner method described in the Modernist Cuisine Vol. 2.

As was pointed out by @RollingCompass, the risk from Botulinum food poisoning is mainly in foods that have not been prepared properly, or improperly stored. It is also quite rare - there were only 19 confirmed cases of food-borne botulinum poisoning in the USA in 2017, which had about 325 million people in that year.

Clostridium botulinum, the bacterial species that produces botulinum toxin, are obligate anaerobes, which means that they will only grow in conditions where there is no oxygen. This is the risk with canning procedures because the canning procedures can lead to exhaustion of any oxygen in the food stuff, but the bacteria can still grow and produce the toxin. Once opened the food is again exposed to oxygen and this will generally prevent growth of Clostridium species. It doesn't get rid of spores or toxins, but I will go into that below.

Assuming that your jar of beans was properly prepared, the spores from the bacteria were inactivated by the high temperatures and pressures during the canning process. In this case, you should be safe. - you would be more at risk of growing other pathogens from storing open food for so long. The Mayo Clinic recommends that if you have home-canned foods, then you should boil for 10 min prior to serving (see section on prevention).

The quote you provided in your question is correct. However, there are two components to the inactivation of botulinum:

1. Inactivation of the toxin - this can be done by heating, as you did, so I won't go into it further.
2. Inactivation of bacteria and their spores.

Spores are incredibly tough, designed to survive a bunch of conditions, including radiation, desiccation and heat. They will survive any boiling process you can put them through in a standard kitchen, except for a pressure-cooker or canner. These raise the pressure of the atmosphere around the liquid and as a result increase the temperature at which the liquid boils. This can result in temperatures above the 121 degrees Celsius that was mentioned.

Placing the lid on an ordinary pot will not increase the pressure inside the pot enough to raise the boiling point of water into the spore inactivation range, so while it might inactivate the toxin and kill the vegetative bacteria, it will not inactivate the spores. However, if you have followed the trendy thing and gotten yourself an Instant Pot or similar device - these fall under the category of a pressure cooker and can be used to sterilize foods.

Temperature of water boiling - 100°C. You boiled it for 15 minutes. So yes, you exceed the time and temperature needed for pasteurisation and denaturation of toxin.

Cindy comment was referring to the fact that botulin need temperature of around 37°C to thrive so a working refrigerator would stop bacteria from producing toxins.

IF and this is very important IF

your white beans in tomato sauce would need to have alive spores. Which, if the beans were store bought is very highly doubtful as typically such food goes through whole time consuming process that kill any type of bacteria in food.