Yes - the shifter has a series of ratchets inside, with one detent per gear, and possibly some additional detents for trim position.
You pushing the lever over is pulling cable, but the pawl is not dropping into a detent and holding itself there. Clearly the cable is moving in your housing successfully, so its not rust in the cable.
The most likely cause on an older shifter is hardened grease blocking the detent, with wear being a lower-prob cause.
A quick fix is to blast the internals of the shifter with solvent to knock down the hardened grease. If that improves shifting for a few months then a teardown may be worth trying, but its fiddly and exact process varies between models.
If you don't intend using the little chainring, it is possible to use the limit screws to force the front derailleur to stay on one chainring. This is inconvenient if you want to ride up any hills though.