Derivatives and the tangent as the "limit of a secant" and as "no other line can be drawn between"

I've read (for example, in Boyer's History of The Calculus) that the definition of tangent as the limit of a secant is due to D'Alembert (1717 – 1783).

So I searched for which other definitions were used, and found that one which says "no straight line can be drawn between a curve and its tangent, from the point of contact, so as not to cut the curve". Doing some search on Google Books I found it seems to be a very old definition too (appearing in texts from MacLaurin, Newton and Lagrange for example).

1st question: How to show these two definitions are equivalent? And how can one use this second definition to find derivatives?

2nd question: Are there other "operational" definitions of tangent which can be also be used to calculate derivatives? (I've heard of "best linear approximation", "locally linear", "zooming", etc)

Sorry if the question is too vague, I really don't know how to write them in a rigorous language.

I'm interested in this because I've seen how to use the "tangent as the limit of a secant with the two points approaching each other", but never saw other methods to solve the same problems.