How to Overlay Text in Premiere Pro or After Effects?
Premiere Pro have more basic text tools, but it is easier to work with. Also, it is better for long files.

After Effects allows you to manipulate with your text in much more various ways. You can use there more different text effects. But it is also more complicated.

So, it is up to you to decide. If you are looking for something basic, use Premiere. If you want to use some fancy effects, then use After Effects.

For example typewriter effect, which is used in video in your question is one of the standard AE effects. You can also achieve that with Premiere Pro, but you will spend more time on each text.