Barometer combines the Greek-derived baro- with a measuring element. The first piece concerns weight or pressure. Weather prediction is an application of that measurement, rather than the literal meaning of the name.
This view explains the components. They are pieces of one construction, not consecutive borrowing stages.
Sources
- Merriam-Webster · barometer
Formation of the word.
- Merriam-Webster · baro-
Meaning and history of this component.
- Merriam-Webster · -meter
Meaning and history of this component.