Let's say we have a mass with an object orbitting with constant speed in a circular orbit and a distant observer Bob. According to Bob's coordinate system, the orbit is circular at a speed v and a constant inward coordinate acceleration a. The coordinate acceleration is just what is inferred according to Bob's coordinate system by the usual definitions of acceleration, s = vo t + 1/2 a t^2, 2 a s = vf^2 - vo^2, a = (vf - vo) / t, etc., where s is the distance travelled and vo and vf are the original and final velocities. The relativistic acceleration formulas also reduce to these when considering a body starting at rest and accelerating an infinitesimal distance over infinitesimal time, which we will be doing, so we need not worry with more complicated formulas for this. So if the object is initially at coordinates x_o=r, y_o=0, and travels an infinitesimal distance y = v t in the tangent direction, it will have travelled a distance of x = r - sqrt(r^2 - y^2) in the radial direction toward the mass, since the orbit is perfectly circular. The distance being infinitesimal, we can drop higher orders and gain just x = r - r (1 - y^2 / (2 r^2)) = y^2 / (2 r) in the radial direction. We want the radial coordinate acceleration as inferred by Bob, so we can use s = vo t + 1/2 a t^2, where vo = 0 for the original velocity in the radial direction, so x = s = 1/2 a t^2. So now we have x = 1/2 a t^2 = y^2 / (2 r), and since y = v t, this reduces to just 1/2 a t^2 = y^2 / (2 r) a t^2 = (v^2 t^2) / r a = v^2 / r giving the usual acceleration formula for a circular orbit. Again, this is just the coordinate acceleration inferred by Bob's coordinate system. It is not saying anything about proper acceleration or anything else, just a coordinate effect. Does all of this look okay so far?