There are some discrepancies between the definition of the Kuznets ratio given at the lecture and in the book (Ch. 6).
In the lectures, we definied the Kuznets ratio as the share of total income received by the richest 20% divided by the share of total income received by the poorest 40%. This is a measure of inequality, i.e. it is a high number if a society is inequal. To see this, notice that the Kuznets ratio is high if the richets 20% receive a large share of income and/or the poorest 40% receive a small share of total income.
The book defines the Kuznets ratio as the share of income received by the poorest 40% divided by the share of the richest 20%. This is a measure of equality, in the sense that it is high if society is quite equal.
Other sources confirm that the definition given in the lecture is closes to what people usually call the Kuznets ratio. As long as you define what you use, you can use the version you prefer.