• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with the following sentence.
In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more.
This is typical a thought experiment which predicts an outcome which can only be validated by a real experiment.
The tricky part is when such an experiment is performed, identical clocks are used. The result is that the moving clock runs behind (which is a very common result with clocks) and the whole explanation is in the mechanics of the clock which uses light signals to count.
This result appears puzzling because each twin sees the other twin as moving, and so, according to an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less.
This prediction, that the clock readings should be the same, is wrong because it assumes a symmetric process, while it is not symmetric.
However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the spacetime paths of the twins.
You don't need SR and you don't need inertial frames (plural). Physics does not care about that. The whole explanation depents about the inner workings of the clocks used.
The twin paradox is called after an experiment. The outcome of any experiment can never be a paradox. The most important issue of an experiment is when you repeat it it must be the same. A different issue is that an observer can try to predict the outcome of an experiment based on similar experiments. When the prediction is correct then our understanding based on the difference in parameters that are relevant.

### 1. History

In his famous paper on special relativity in 1905, Albert Einstein deduced that when two clocks were brought together and synchronized, and then one was moved away and brought back, the clock which had undergone the traveling would be found to be lagging behind the clock which had stayed put.
All of this depents about the innerworkings of the clock.
The whole idea is that when you move a clock the speed of light does not change and stays constant. A standard clock works based on something that vibrates or pulsates. That can be a light which moves back and forward between two plates. When you move such a clock in any direction the length of the path travelled will be longer and the clock will tick slower.
Einstein considered this to be a natural consequence of special relativity, not a paradox as some suggested, and in 1911, he restated and elaborated on this result as follows (with physicist Robert Resnick's comments following Einstein's):
If we placed a living organism in a box ... etc
Every experiment where humans are involved in the experiment is not very scientific.

### 2 Specific example

Consider a space ship traveling from Earth to the nearest star system: a distance d = 4 light years away, at a speed v = 0.8c (i.e., 80 percent of the speed of light).
How do you know that the speed is exactly 0.8 times the speed of light in this experiment?

### 3.1 Role of acceleration

Although some solutions attribute a crucial role to the acceleration of the travelling twin at the time of the turnaround,others note that the effect also arises if one imagines separate outward-going and inward-coming travellers, who pass each other and synchronize their clocks at the point corresponding to "turnaround" of a single traveller.
When you perform experiments with different speeds always accelerations are involved.
The whole explanation is again that the speed of light (of photons) is a physical constant process. If you move a clock the relatif speed will change which causes a change in the performance of the clock. That is all
In this version, physical acceleration of the travelling clock plays no direct role; "the issue is how long the world-lines are, not how bent".
I do not think that the concept of worldlines is required.
In Minkowski spacetime, the travelling twin must feel a different history of accelerations from the earthbound twin, even if this just means accelerations of the same size separated by different amounts of time, however "even this role for acceleration can be eliminated in formulations of the twin paradox in curved spacetime, where the twins can fall freely along space-time geodesics between meetings".
This explanation is not clear and difficult to understand.

### 3.2 Relativity of simultaneity

For a moment-by-moment understanding of how the time difference between the twins unfolds, one must understand that in special relativity there is no concept of absolute present.
In classical mechanics generally speaking only one reference frame is used. The position of the objects considered in that reference frame can change in time.
For different inertial frames there are different sets of events that are simultaneous in that frame.
That seems very logical.
This relativity of simultaneity means that switching from one inertial frame to another requires an adjustment in what slice through spacetime counts as the "present".
That is correct. In fact for the moving twin there are three inertial frames involved.

### 12 No twin paradox in an absolute frame of reference

Einstein's conclusion of an actual difference in registered clock times (or aging) between reunited parties caused Paul Langevin to posit an actual, albeit experimentally undetectable, absolute frame of reference:
The whole issue is that in SR in every inertial reference frame the speed of light in any direction is c.
In classical mechanics only one reference frame is considered in which this is true. The behavior of a moving clock as such will be different based on a clock at rest in that frame.
In the relativity of Poincaré and Hendrik Lorentz, which assumes an absolute (though experimentally indiscernable) frame of reference, no twin paradox arises due to the fact that clock slowing (along with length contraction and velocity) is regarded as an actuality, hence the actual time differential between the reunited clocks.
It is difficult to understand what they mean. The point is that clock slowing is an actuality and can be observed by comparing the readings of the two clocks. Length contraction is not an actuality because it cannot be observed.