I'm unaware of any definition of 'vaccine' which fits seasonal flu shots, but not COVID vaccines. Specifically, flu vaccine does not prevent infection or transmission, it has moderate therapeutic benefit and reduces instances of infection somewhat. That meets your presented criteria in full.

You're correct that many aspects of how the vaccine was handled in terms of authoritarian social policy, overpromising on results, sweeping bad reactions under the rug, and much more, has undermined public trust in vaccination.

But this has no bearing on whether the half-dozen or so vaccines developed against the virus are vaccines. They are.