IMHO, Euclid's definition of a straight line in today's terms would be "a line that has the same direction on its entire length". His definition of a plane angle would be "a plane angle is the difference between the directions of two straight lines that have a common end in one point".
What are Playfair's and Hilbert's definitions?
Playfair interprets Euclid as follows, I am using my own words here, a straight line is that figure which has the property that if it intersects its moved copy at 2 points it necessarily coincides with it everywhere. "Movement" is undefined, it has to be an isometry.
Hilbert's is more abstract and based upon sets. Line is a primitive (undefined name) that interacts with two other undefined names (points and planes) according defined relations (lies on, lies between and is_congruent).