How do I record a weightlifting exercise on an Apple Watch?
As far as I can tell, your best bet is to go through this process:


On your Apple Watch open the Workout app
Use the crown to navigate to the Other option
Tap on this to begin your weightlifting exercise workout
After completing the workout, swipe right and tap on the red END cross
When the summary appears you'll see a grey Name Workout button - tap on that
Now you'll see a list of categories, initially grouped by Popular categories and then listed in alphabetical order
Use the crown to scroll through the choices and tap the one that best fits (I see that Strength Training is listed as a popular choice - perhaps that's a suitable option)
Now the label will appear on the Summary screen
Use the crown to scroll to the bottom of the summary screen and tap on the Done button


Now you'll find that whatever category you selected will be available in your list of main Workout options when you launch the Workout app. It also means this option can be used by the Activity app on your iPhone so that your workout history is more informative/applicable to what you're doing.