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

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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.