You don't need to record the weights you lift. You can tell how hard something is to lift just by picking it up. I lift weights all the time, and I couldn't tell you how much I could lift if I wanted to. I set a machine to a hard amount to lift, and I just lift it a bunch of times until I can't lift anymore.
I was totally out of shape two years ago and started doing both lifting and running. My progress was quicker with lifting then running so I've chosen to focus more on lifting. I like the Immediate feedback of getting one more rep then previously instead of having to finish a run and then see my time. ALthough my cardio is not good by any means lifting three times a week and running two times a week has made me feel more energetic in my daily life. Since I have no desire to do races that's good enough for me. This may change now that i have a Xiaomi mi band, I've found my self spending more time on the Treadmill even if it's just walking so I can get that vibration telling me I hit my goal for the day.
I'm a sprinter, so I do take a quantitative approach to running.
I also lift weights, and I would say that I do so seriously. But my weights are my own body and big rice sacks filled with water bottles that I heave around in various ways until I'm tired. I'm an experienced enough athlete to know when I'm hitting my limits, without following any specific metric.
With lifting weights, you lift a specified fixed numerical quantity that generally you need to record so you lift the same or more the week after.