In most studies, caffeine intake was associated with no significant weight change or a slight weight loss.
Caffeine - no weight change
1) Caffeinated Coffee Does Not Acutely Affect Energy Intake, Appetite, or Inflammation but Prevents Serum Cortisol Concentrations from Falling in Healthy Men (The Journal of Nutrition, 2011)
Sixteen apparently healthy, nonobese, young men who were habitual coffee drinkers were recruited by local advertisement to participate in this study.
In conclusion, the usually consumed amount of caffeinated coffee does
not have short-term effects on appetite, energy intake, glucose
2) Weight gain in older adolescent females: the internet, sleep, coffee, and alcohol (The Journal of Pediatrics, 2008):
A longitudinal cohort of >5000 girls (Growing Up Today Study), from
all over the United States and aged 14 to 21 years, returned surveys
in 2001 reporting typical past-year recreational Internet time, sleep,
coffee (with caffeine), and alcohol consumption.
We found no evidence that drinking coffee promotes weight gain.
Caffeine - weight loss
1) The effects of caffeine intake on weight loss: a systematic review and dos-response meta-analysis of randomized controlled trials (Critical Reviews in Food Sciences and Nutrition, 2019)
Thirteen randomized clinical trials with 606 participants were included in the
Overall, the current meta-analysis demonstrated that caffeine intake
might promote weight, BMI and body fat reduction.
2) Body weight loss and weight maintenance in relation to habitual caffeine intake and green tea supplementation (Obesity research, 2005)
A randomized placebo‐controlled double blind parallel trial in 76 overweight and moderately obese subjects (BMI, 27.5 ± 2.7 kg/m2)
High caffeine intake was associated with weight loss through
thermogenesis and fat oxidation and with suppressed leptin in women.
In habitual low caffeine consumers, the green tea-caffeine mixture
improved WM, partly through thermogenesis and fat oxidation.
This claim from the source in the question:
Constant activation of the body's fight or flight system (via the
daily ingestion of caffeine poison) aids in a metabolic shift to fat
storage and fat conservation, because again the body prefers fat as a
fuel source when fighting any toxic intruder…
...is contradictory: If the body's primary fuel when activated is fat, how does this result in fat conservation?
Caffeine does not promote fat storage but its degradation (oxidation):
In conclusion caffeine/coffee stimulates the metabolic rate in both
control and obese individuals; however, this is accompanied by greater
oxidation of fat in normal weight subjects. (AJCN, 1980)
In conclusion, the summary of evidence does not show that caffeine/coffee stimulates weight gain, but it also does not convincingly show that it stimulates weight loss.