I don't know what at level you are studying, but your description suggests you are confusing what are known as the steady state and transient solutions to equation of forced vibrations. Does this mean anything to you? no it does not really! I am at AS Level Physics - Edexce...
There is therefore no phase change in this reflection BUT a standing wave is still produced? Don't forget that the animation is of an ideal string. An ideal string cannot generate standing waves if it has a free end. Try to get hold of the concepts first, but keep going...