It can do both. You describe the NFC key (NFC = Near Field Communication). This works just like your physical key card. Using this key, you must unlock and lock the vehicle "manually" on the b-pillar. With NFC, the watch is not connected to the vehicle and therefore cannot automatically lock the vehicle when you leave. At no point should it be neccessary to take your watch off to put it on the card reader. After tapping the b-pillar to unlock, you will have quite a long time to drive off without needing to reauthorize behind the cupholder. And if you need to, you only need to tap the watch, just like you do on the b-pillar.
But you can also pair the watch as a Bluetooth key. Then it will work the way you want it to by connecting using Bluetooth. After you have paired the app with the car, you must choose which power mode you want to use. Default mode, to save the watch's battery, is that you have to open the app or swipe to the Tile to connect the Bluetooth key. Another power mode is to shake the watch to connect (screen must be lit before shake). The third power mode is that Bluetooth is always on. This power mode will affect the watch's battery usage since the watch will constantly search for the car 24/7. Choose power mode by tapping the battery/heart icon in the Tesla key settings found in the app settings menu.
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PS! If you have previously added the watch as a NFC key from the car's UI, you must remove this key from the car's UI before you re-pair the app as a Bluetooth key. As soon as you have paired the Bluetooth key, it will work both as an NFC key and as a Bluetooth key.