Its hard to find adequate diagrams to illustrate this, but I'll explain as best I can.
For the average person, the teaching to not "lift with the back" is good advice
Someone who isn't used to lifting what the average person would consider heavy loads (lets say a 20kg box), bending at the waist to lift causes a rounding spine. Combined with weak hamstrings and back muscles, this rounding leads to unnecessary strain on the spine and can cause disc damage.
Instead, people are taught to stand close to the weight, pull it close and lift from a squat position, using the stronger quad muscles. Lifting from this position helps keeps the back straight and minimises pressure on the discs of the spine.
But, lifting with your back is not necessarily dangerous
As you pointed out, there are plenty of exercises that use the lower back muscles, such as extensions and deadlifts. However, you'll notice whenever people advocate for these exercises they stress the importance of good form - much more that in most other cases. Once trained in or aware of how to straighten and strengthen the spine, and brace with the "core" muscles to increase Intra-abdominal pressure, deadlifts are perfectly safe. Thus, for someone who has a 100kg deadlift, who keeps good form, straight leg lifting a 20kg box from the floor is perfectly safe.
What does this mean for you?
Firstly, always keep a strong/straight spine when lifting anything that isn't a balanced and uniform barbell. One difference between weightlifting and what is taught as "safe-workplace lifting" is the load is usually uniform and easy to lift in the gym, whereas the a box in the workplace may be imbalanced and hard to hold. So good spinal form is the most important part of the lift.
Secondly, continue to strength your lower back. With good form there is absolutely no danger to training your lower back like any other body part.