The second one, at least, is wrong in two ways: The vaccines development was focused on reducing symptoms, and we don't know the limits of how long vaccine or natural immunity will last, simply because neither has been around that long.
Last I remember reading, both are "for as long as it's been studied so far" - 8 months for natural immunity, 3 months for the vaccines. These numbers will increase as we continue keeping track of it to see how much the effect degrades over time.
Firstly, we have a lot more than 3 months of data now. This study of the Moderna mRNA vaccine shows that after 209 days "antibody activity remained high in all age groups" and and "nearly all participants had detectable activity in a pseudovirus neutralization assay".[1]
Secondly, sure, we don't know how long vaccine immunity will last, but that's a far cry from saying that vaccines were not designed for lasting immune reponses. For example, Pfizer/NTech's phase I/II trials explored over different doses, and they ended up selecting a dose of 30 micrograms rather 10 micrograms or 1 microgram because 30 micrograms elicited greater antibody responses, sustained over their measurement period of 43 days.[2]
Thirdly, quoting from the abstract of that same Nature paper: "Two doses of 1–50 μg of BNT162b1 elicited robust CD4+ and CD8+ T cell responses and strong antibody responses, with RBD-binding IgG concentrations clearly above those seen in serum from a cohort of individuals who had recovered from COVID-19." [2]
Let me state it again, without the jargon: Vaccinated patients had antibody responses clearly above those who had recovered from COVID-19. This, as I understand it, is why the CDC recommends that those who have recovered from COVID-19 still get the vaccine, as it offers extra protection from weaker or faded immune responses. [3]
Do you really think my statement that 'the vaccine is specifically designed to trigger strong and lasting immune responses' is wrong?
As I shared above, the phase I/II trials of these vaccines measured time series of immune responses to help select the dosing, and they picked a dose that had a strong immune response that lasted over their measurement time series.
Last I remember reading, both are "for as long as it's been studied so far" - 8 months for natural immunity, 3 months for the vaccines. These numbers will increase as we continue keeping track of it to see how much the effect degrades over time.