The thing to get in understanding Buddhist literature is translation is an art, and when dealing with complex philosophies the translator must also live the lifestyle to understand it.
So, I agree that the Pali term 'sati' is reasonably translated as 'mindfulness,' but we have to understand what mindfulness is, especially as it was understood by Siddhartha Gautama, who arguably had a more nuanced understanding of it than popularly held beliefs in his time and culture.
So, when I use the term 'mindfulness' I also add in 'self-awareness' as a term used in western contemplative traditions with a subtle nuance that non-contemplatives will never understand. To be mindfully self-aware is to observe one's self at high resolution beyond normal non-mindful population. This means translator are almost never contemplatives. And, priests of all religions, including Buddhism are rarely contemplatives. In all religions priests more often than not pretend to be holy and put nf the religious garb and behavior believing if they pretend well enough and long enough they will eventually become holy, but instead what we see is corruption and pedophilia.
So, being mindfully self-aware is to observe yourself in deep detail, and the 4 types of mindfulness are really more like 4 levels of mindfulness, body, breath, feelings, thoughts and dhamma.
Now we have to define the Pale and Sanskrit terms 'dhamma.' Dhamma is used in a number of ways in the Pali canon. It surely means the philosophy that Siddhartha Gautama taught. However, it also has the subtle nuance of belief systems in general. So, as I interpret the Pali term 'sati-dhamma' is in modern terminology is 'unpacking one's belief systems. We cannot expect to embrace contemplative philosophy as deep as the philosophy of Siddhartha Gautama without thoroughly unpacking our belief systems.
For instance we western people have been influenced by the western religions which are based upon the imported middle eastern Abrahamic religions. Consequently they are deeply invested in a belief in a benevolent creator-god who materially punishes the 'wicked' while materially benefiting the 'righteous.' When we unpack this philosophy we find it was developed and advanced to benefit the wealthy who were more often than not unrighteous.
Another deeply flawed western philosophy is all we have to do is believe something and we will gain both material and spiritual benefit. This deeply flawed belief system has been at the center of organized religion's attack on western contemplative movements as we have discussed elsewhere and evidenced by the Arian heresy and the attack on witchcraft and the quietist movement in the 17th century.
So, be mindfully self-aware at all levels and unpack your belief systems that undermine your success in the contemplative arts.