I think the existing answers, at this time, all raise important points, and I have nothing to add to the ones already made. However, I feel it is worth emphasizing that, when teaching, it is important to bear in mind the ultimate purpose of the education, and not “make the material fit our teaching” but “make our teaching fit the material.”
There is simply no question that, if you are teaching students who are studying (or will go on to study) physics, engineering, math or any other subject that uses applied mathematics, they would be quite literally “uneducated” if they did not understand the difference (or lack thereof) between “Power Series,” “Taylor Series” and “Maclaurin Series.” It is simply part of the language and culture of these fields, even if confusing when first learning, and/or inconvenient when teaching.
Of course, if the students will not go on to use these tools and the content is covered in a course that marks the end of their studies, then it is surely less important (for the reasons covered above).
A related issue is the language used to express the centre of convergence: “centred at $x = a$” and “in powers of $(x-a)$ are both equivalent expressions that, for better or worse, students who will use these ideas in the future need to know.
In short, the language really does matter for some students; however, the distinctions are easily introduced once a firm understanding of the ideas exists (much as the difference between different breeds of dogs is easily introduced once one has a firm grasp of “dogs” as a group, first!); in fact, confusion in the terminology can be used to diagnose a lack of understanding of the basic ideas and can, in that way, be turned into a pedagogical advantage.
On the practical considerations associated with teaching these topics:
I have found the most success (in avoiding confusion surrounding the language used) by introducing the idea of a power series, developing the related idea of power-series representations, and then simply mention, as an historical / cultural note, that “Taylor/Maclaurin series” is commonly used instead of “power series representation,” once students are comfortable with the ideas. Similarly, I have found that students are better equipped to understand the connection with “tangent lines” and “linear approximation,” after they have grown comfortable with the definitions and computations involved in power series (representations).
I’m not sure if the phenomenon is widespread, but students in my neck of the woods generally seem to have a very poor understanding of power series and power series representations (reflecting an equally poor understanding of tangent lines and linear approximation), in general, so focusing on clearing up these points usually makes the later clarification of terminology a non-issue.
As suggested by others in the comments and other answers, I have found that emphasizing power-series expansions as an extension of the tangent line is fundamental to helping students understand (again, from an applied-maths perspective, I would want all my engineering and science students to know that “first-order Taylor series expansion,” “tangent line” and “(best) linear approximation” are all equivalent expressions).
Finally, a strong emphasis of graphical presentation of the material always seems to be essential for most students.