There is a type in Welch test degree of freedom calculation. Nominator's square exponential is in another place, please fix it.

Thank you for your input, we we'll clarify this.

In your data frame, women's weight comes first:
weight = c(women_weight, men_weight)

And you wrote:
Note that: if you want to test whether the average men’s weight is LESS than the average women’s weight, type this:
t.test(weight ~ group, data = my_data, var.equal = TRUE, alternative = "less")

You implied:
if you want to test whether the average women’s weight is GREATER than the average men’s weight, type this:
t.test(weight ~ group, data = my_data, var.equal = TRUE, alternative = "less")

But, let's see the documentation via ?t.test:
t.test(x, y.....)
alternative = "greater" is the alternative that x has a LARGER mean than y

Hence, you should change like this:
Note that: if you want to test whether the average men’s weight is LESS than the average women’s weight (i.e., the average women’s weight is GREATER than the average men’s weight), type this:
t.test(weight ~ group, data = my_data, var.equal = TRUE, alternative = "greater")

And similarly you should change for the womens' LESS case as well.