As of the movement itself, this can be the outcome of several mindsets and tactics. Therefore, I will use the Japanese terms that describe a certain tactics/mindset while fighting rather than the relative movements of the opponents as they appear.
In contemporary Japanese martial arts, it is "sen" (or "sen no sen").
From Tadao Otaki & Donn F. Draeger (2001) Formal Techniques: A Complete Guide to Kodokan Randori no Kata, Tokyo: Tuttling, p.78:
Kano recognized three levels of combative initiative (sen): ( I) go no sen, the "late" form
of attack initiative, usually characterized as a defensive move or counteraction; (2) sen,
the attack initiative that is also defensive but launched simultaneously with the aggressor's
attack; (3) sen-sen no sen, a supraliminal attack initiative, also defensive but appearing
to be offensive, through which the aggressor's attack is anticipated and "beaten to
the punch" by an appropriate action.
For the variant "sen-no-sen", see Lowry, Dave (2017): Traditions: Essays on the Japanese Martial Arts and Ways, Tokyo: Tuttling, chapter 36 ("Sen (Taking the Initiative)").
In older Japanese (military tactics), it is "tai no sen"
This is hard to apply as it is about troop movements and actual attack movements of opposing armies instead of 1 on 1, hand to hand fights. This becomes apparent in the longer descriptions of the principles given, especially in the German edition that used the original manuscript to correct the English version (and which I cannot cite here for obvious reasons). Thus, these terms and descriptions are to be taken metaphorically, not literally.
From the classic The Book of Five Rings by
Miyamoto Musashi (1645):
The Three Methods to Forestall the Enemy
The first is to forestall him by attacking. This is called Ken No Sen (to set him up).
Another method is to forestall him as he attacks. This is called Tai No Sen (to wait for the initiative).
The other method is when you and the enemy attack together. This is called Tai Tai No Sen (to accompany
him and forestall him).
There are no methods of taking the lead other than these three. Because you can win quickly by taking the
lead, it is one of the most important things in strategy. There are several things involved in taking the lead.
You must make the best of the situation, see through the enemy's spirit so that you grasp his strategy and
defeat him. It is impossible to write about this in detail.
Remark: I'd say it is rather "tai no sen" than "tai tai no sen" because it is about dodge and attack in one move, i.e. the initiative of the opponent is part of my course of action through which I "forestall" his initiative.
There are ambiguities that may not even be visible to the untrained eye, i.e. a dodge can be preemptive (reacting on telegraphing) coupled with a counter or an adaption of a striking initiative that was there before (both attacking at the same time), which may make the difference between tai no sen and tai tai no sen and look the same.
The longer descriptions given are:
When the enemy attacks, remain undisturbed but feign weakness. As the enemy reaches you, suddenly
move away indicating that you intend to jump aside, then dash in attacking strongly as soon as you see the
enemy relax. This is one way.
Or, as the enemy attacks, attack more strongly, taking advantage of the resulting disorder in his timing to
This is the Tai No Sen principle.
When the enemy makes a quick attack, you must attack strongly and calmly, aim for his weak point as he
draws near, and strongly defeat him.
Or, if the enemy attacks calmly, you must observe his movement and, with your body rather floating, join
in with his movements as he draws near. Move quickly and cut him strongly.
This is Tai Tai No Sen