Cox-Stuart vs Augmented Dickey Fuller

Given a time series, suppose I wish to determine if trend is present. I understand I can use the Cox-Stuart test (https://www.r-bloggers.com/trend-analysis-with-the-cox-stuart-test-in-r/), but I have also read that the Augmented Dickey Fuller test can be done for this (https://machinelearningmastery.com/time-series-data-stationary-python/). More specifically, the last link says:

"The Augmented Dickey-Fuller test is a type of statistical test called a unit root test.The intuition behind a unit root test is that it determines how strongly a time series is defined by a trend."

But I have also read from (https://kourentzes.com/forecasting/2017/03/30/can-you-spot-a-trend-in-a-time-series/):

"I do not think it is an exaggeration to suggest that even experts often can be confused on the exact definition (and effect) of a unit root and a trend."

So my questions are thus:

1. What does it mean for a time series to have a unit root? Does that automatically mean there is a trend in the time series?
2. What is the difference between the Cox-Stuart and Augmented Dickey Fuller Tests? Which should I definitely use for trend detection?