There are two different "rates" to consider, and there may be a language issue. There is the sample rate which in your code is 1Hz as determined by the delay(1000) between readings. Then there is the internal conversion rate of the ADC which is controlled by the ADCSRA bits which you are changing from the default settings of the Arduino IDE.
How do you know that the conversion rate was not changed?
Nick Gammons excellent tutorial on the Arduino ADC may be of interest
https://www.gammon.com.au/adc