Differentiation help - points of inflection

I'm a bit confused. I understand that to find non-stationary points of inflection, we find the points on the curve where the second derivative = 0 and check the second derivative either side of those points (checking for concavity), which makes sense. But to find 𝘀𝘁𝗮𝘁𝗶𝗼𝗻𝗮𝗿𝘆 points of inflection I have been told that once we have found the stationary points, and we know that d^2y/dx^2 = 0 at that point, then we only need to check that the 𝗴𝗿𝗮𝗱𝗶𝗲𝗻𝘁 is the same either side of the stationary point to be able to conclude that it is a point of inflection. Why do we not need to check for concavity? Are there no other possible natures of a stationary point other than maximum, minimum or point of inflection?

I'm a bit confused. I understand that to find non-stationary points of inflection, we find the points on the curve where the second derivative = 0 and check the second derivative either side of those points (checking for concavity), which makes sense. But to find 𝘀𝘁𝗮𝘁𝗶𝗼𝗻𝗮𝗿𝘆 points of inflection I have been told that once we have found the stationary points, and we know that d^2y/dx^2 = 0 at that point, then we only need to check that the 𝗴𝗿𝗮𝗱𝗶𝗲𝗻𝘁 is the same either side of the stationary point to be able to conclude that it is a point of inflection. Why do we not need to check for concavity? Are there no other possible natures of a stationary point other than maximum, minimum or point of inflection?

It would probably help to see the exact definitions youre talking about, but a point of inflection is when the second derivative changes sign, so you usually check where the second derivative equals zero then verify that it changes sign in the local neighbourhood. Just like to find a turning point, you find the stationary points (gradient is zero) and verify that the gradient localy changes sign.
But you seem to be asking about if you know that a point has zero first and second derivative and the derivative is the same sign either side of that point. So the gradient curve would locally be like a "u" or a "n", then that would correspond to a turning point of the gradient and the second derivative must change sign.
Its basically the same as saying that a curve has a turning point if the function and the gradient is zero at a point and the function locally has the same sign.

Essentially what mqb said, but in slightly different words, a turning point (max or min) is essentially where the gradient turns - imagine sliding a ruler along your curve (let's say y = x^2) so that it's just touching (tangent to) the curve, then it might start by sloping downwards (northwest to southeast) then level off - that's your turning point - then start to rise upwards (southwest to northeast). The point where it "levels off" is your max or min. Now this is also a "stationary point" - defined as where dy/dx = 0 - but not all stationary points are turning points - you can have an instance like y = x^3 where the gradient is instantaneously 0 (at x = 0) but the gradient is +ve on either side of this point. If this is the case, then you have a (stationary) point of inflection.
Not sure if this helps, but essentially knowing how the gradient behaves removes the need for a separate check for concavity.