I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say will be valid. go for it!
However it is worth noting that there is a lot of historical precedent both for teaching it algebraically, as is done in algebraic geometry, i.e. via the concept of orders of vanishing, but also as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). Thus the tangent is the limit of those secants. Thus one could say that the limiting point of view is the original one of Euclid.
On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for derivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2.
Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root.
But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent.
If you want a defense of making students practice using the limit definition, I propose that this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that they have not grasped either what an abstractly defined function is, nor what a derivative truly means. So if you want them to understand the relation between the derivative and the integral, then they need to know what a derivative is.