In the physics applications that Newton was interested in, his functions were mostly functions of time. Since he was usually differentiating with respect to time, it was OK that his dot notation, unlike Leibniz's notation, didn't indicate what he was differentiating with respect to. Newton used the symbol $o$ to indicate a fixed infinitesimal interval of time. So in Newton's notation $\dot{x}$ would be Leibniz's $dx/dt$, while $\dot{x}o$ would be $dx$ --- an infinitesimal difference, called a "moment."

Newton also used a notational shortcut that caused confusion. He used a convention that when it was clear from context that he was talking about a moment, then $\dot{x}$ would implicitly mean $\dot{x}o$, i.e., the $o$ could be left out because it was too cumbersome to write it all the time. Because of this, English mathematicians began to muddy the notational waters by not distinguishing the two notions. See Boyer, p. 201, and p. 114 for the definition of "moment."

So when someone using Newton's notation says that the fluxion of $x^n$ is $nx^{n-1}\dot{x}$, what they mean could be two different but equivalent things in Leibniz notation. Either:

(1) $d(x^n)=nx^{n-1}dx$ (where $dx$ is the same as $\dot{x}o$, and the $o$ is implicit); or

(2) $d(x^n)/dt=nx^{n-1}dx/dt$

In 17th- and 18th-century mathematics, the difference between (1) and (2) is purely a matter of dividing both sides of the equation by $dt$. (There was no notion, as in modern NSA, that a derivative is the standard part of the ratio of infinitesimals, which is not quite the same thing as the ratio of infinitesimals.)

Boyer, The History of the Calculus and its Conceptual Development. https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment