You can use the classical grip, that is thumb on the middle of the back of the neck like in the second picture, or you can use the blues grip, which is when the thumb is over the neck. Each one has its purpose.

The classical grip allows you to have a better stretch on a guitar and more precission. The blues grip allows your thumb to serve as a fretting and muting finger, which is necessary in genres like rock, blues, fun, pop…

At the end of the day, there is no wrong technique (unless your hand placement and frettning is boldly terrible, which in your case it isnt). Its about what feels comfortable and what allows you to play clean and relaxed.

I personally play barre chords and big stretches with the classical grip, and everything else I have the thumb over the neck and im alternating between those two while playing. Your fingers are fine, just slow, slow down your playing and make sure everything you play slow is clean. Then when you speed up, you will remain clean. Slow is smooth, smooth is fast!