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The stay-at -home twin is at rest in her frame and her clock must therefore measure proper time.Also note that in the twin paradox we are comparing one person's proper time with another person's proper time. It is not the same thing as time dilation, and not understanding that can be a stumbling block to understanding the twin paradox. Many if not most published explanations of the twin paradox fail to mention this, and in my opinion that can cause confusion for the reader who is trying to understand it.

The travelling twin, carries his clock with him; it is therefore at rest in his frame and must also measure proper time.

As each twin is moving relative to the other, they will each measure coordinate time for their twin.

Their proper times will be identical.

Their coordinate times will be identical.

As their relative speeds are the same, their Lorentz transformations will be the same.

When the travelling twin slows on his return and comes to rest in his twin's frame they are both once again in the same frame and will have travelled exactly the same each relative to the other.

It is only if the traveller continues past at speed that there is a time difference as they pass and that will be proper time vs coordinate time...