Sophia A common feature on the original electric pianos, like Fender Rhodes, Wurlitzer 200, was a tremolo, similar to that on organs. Later on, when stereo was available, this tremolo could be oscillated from left to right to left, on and on, and with headphones it's very noticeable. This effect is called panning. Without seeing the e373, I would guess that preset has that effect turned on, but it's probably adjustable as to how much, speed, or even turn off.
Your Clavinova doesn't have a headphone jack? I thought even the earliest models had one. If you discover that it does, that can be used to record audio. But, you'd still need that "stone aged" audio input on your phone, or whatever your recording on.
By the way... most modern phones have pretty good microphones. So, nothing to do with the little speakers on the phone. My relatively old iphone SE has a microphone good enough to record bird songs from around our yard with surprising clarity. And, it also has the 3.5mm headphone/mic jack, which is one of the reason I still use it.