Wow. Relativity isn't relative to observers, and somehow a theory in which there exists no absolute time or simultaneity somehow allows for "time" (which isn't treated independently of space in the theory) itself to "speed up." Ok, two questions:The affect would exist from a third observer that would see one object slowing down and another object speeding up time keeping it relativistic. It isn't relative to an observer but relative to amount of mass.
1) Is there any source anywhere that describes what you are talking about and is at least as authoritative as Wikipedia? Clearly, either you are misunderstanding some source or misinterpreting some use of terminology.
2) Can you explain your understanding of proper time in special relativity?
To help:
"This effect is known as time dilation: the time interval between two events depends on the state of motion of the observer. The process in the (x, t) plane is illustrated in Fig. 1.12. We define the proper time interval between two events as the time interval measured by an observer who sees the two events occurring at the same point of space." (emphasis added)
Faraoni, V. (2013). Special relativity. Springer.
"This is the time dilation effect; the time between the two events depends on the observer. It is paradoxical only if one insists on thinking about time independently of the process of measurement of time."
Woodhouse, N. M. J. (2014). Relativity Today. In A. Ashtekar & V. Petkov (Eds.) Springer Handbook of Spacetime (pp. 39-60). Springer.