Nearly correct.
"DTR line goes high and the capacitor charging slows"
When one end of a capacitor suddenly changes voltage, the other end of the capacitor instantly follows that change.
So a capacitor with both ends at ground has no charge in it. If one end goes to 3V3, then so does the other end, because the charge on the capacitor (in this case no charge) has to be maintained. This causes a hig voltage on the reset pin which causes the reset.
But now there is a voltage across the capacitor and so it starts to charge up, which causes previously grounded end, which has jumped to a high voltage to start to drop, thus releasing the reset.