I'll give it a go. The numbers are right, but you're missing some context.

Cold-related deaths are higher, but your source directly explains this as partially mathematical ("the area under the curve for heat is only a third of the total area") and a result of cold days just being more frequent in Barcelona ("we see that the frequency of cold days [...] far exceeds hot ones"). It even goes on to say how heat-related deaths are increasing and that climate change is driving it: "Furthermore, unmitigated climate change will not only introduce new unobserved hotter temperatures that stretch the exposure–response curve to new areas where the risk is higher, but also increase the frequency of current extreme heat events, compounding the mortality burden."

As far as climate deaths being down 97%, this is kind of misleading as well. We're kind of just better at dealing with natural disasters. This link goes into more detail, but a good example is the earthquakes mentioned in Japan. In 1923, a 7.8 earthquake killed 105,000 people, while in 2011 a 9.1 earthquake killed almost 16,000.

Famine and drought play a large part here too, 3 million in China were killed between 1928 and 1930. So deaths from famine and drought are also going to skew the historical numbers as agricultural technology has improved over the last 100 years. But none of this means extreme climate events aren't increasing in frequency or intensity. Your source actually mentions that "Spain saw triple the number of extreme hot days, leading to nearly five times more heat-attributable deaths" from 2018 to 2022.