The short answer is "yes", ITF and Kukkiwon both have them; but the long answer is more pedantic (and still "yes").
- ITF has "Yopcha Milgi"
- Kukkiwon has "Mireo Chagi"
I can't answer for ATA or any other TKD style
From a purely physics POV, there is no difference between a push, thrust, or pierce (or even a block and parry...) All involve a movement along a vector, and the thing moving is met with an object along that vector. The name of the technique usually boils down to intent.
We may qualify the action moving along the vector as a kick when the thing kicking is moving at a fast rate of speed; and similarly we may qualify the action as a push that thing doing the action is slow. (With regards to hands or body motion, we might even discern the difference between a push and throw having more to do with velocity than speed, but that is outside the scope of your question).
We can also take a look at some technique definitions. In Choi's encyclopedia, Vol II, beginning page 76, nowhere is speed of the kick mentioned. It only mentions optimal attacking areas, and parts of the foot. But speed is not mentioned at all. In fact, on page 76, it specifically states "Proper and timely employment of this tool [the foot parts] cannot be overemphasized". Key word here is "timely", which is not the same as speed or velocity.
In Vol IV, pages 25, 36, and 38 describe techniques "Yopcha Jirugi" (side piercing kick), "Yopcha Tulgi" (side thrusting kick), and "Yopcha Milgi" (side pushing kick), respectively. The relevant differences are described here:
YJ: No mention of speed is made at all. Strike only with the footsword.
YT: variant of YJ, strike only with ball of foot.
YM: variant of YJ, uses only weight without acceleration and power, losing piercing force and naturally rapid withdrawl of foot becomes less important. No mention about whether to use the ball of toes, the heel, or the foot blade. It DOES say "be sure to cross the other foot rapidly past where the kicking foot had been placed while KICKING AND PUSH the target momentarily". Also, "without acceleration" is interesting, as it suggests the others should be using acceleration, whereas here there is no acceleration. Gen Choi did not seem to have a good grasp of physics when considering all other discussions of power, and so, one might question whether he really meant "slow speed" when he said "no acceleration", because, a constant velocity has an acceleration of 0, even if you have an extremely fast kick.
So as far as ITF is concerned, the implication is that velocity of the kick is important as far as push vs thrust/pierce is concerned, but, seems not well defined. This suggests that in ITF, all kicks are both pushing and piercing/thrusting. (Note that a thrust and a pierce are exactly the same, except the thrust uses the ball of toes, while the pierce uses the blade. The push has "no acceleration".)
In the Kukkiwon style of Taekwondo, the definitions according to www.kukkiwon.or.kr (and similarly described in its textbook) are much more vague:
Mireo-chagi is a "push kick" which applies to the front and side kicks, and is described as "A technique of pushing the opponent with the foot. This is a pushing technique to make the opponent fall or to maintain some distance with him or her by using the ball or sole of the foot." Note here the technical difference: ITF doesn't say ball/toe/blade/sole, while KKW says ball/sole can be used.
Yeop-chagi (side kick) is a "thrusting kick", which is described as "A technique of delivering a kick by turning one’s body to the side. This is a skill of striking the opponent’s face or chest with the outside edge of the foot or the bottom of the foot. The side kick is usually performed by turning one’s body to the side and at the same time bending the leg and extending it to kick the target. In some circumstances, one may push the opponent with the foot to maintain the distance from the opponent."
As I said, it's pedantic. But it means all kicks can be "piercing", "thrusting", and "pushing". It depends on your application, which should be part of your poomsae / hyung. And one's definition of "push".
EDIT: with apologies, I referenced the wrong volume, should be IV (4), not IX (9), pages 25-38.