“These cause the cells in your body to produce something for your body to attack - it’s not an external threat that’s being killed, it’s something your own body produced. “
That is correct. But these cells sprouting the spike proteins die (killed by the body’s own immune system). The key point is they are limited in number — and all of which will die — and the change is not hereditary. The spikes won’t be passed down to succeeding generations because there is no genetic/DNA alteration.
mRNA has a short half-life (average half-life of mRNA has been shown in some studies to be ~4.8 minutes). Once you've been injected the mRNA encoding the spike protein will be taken up into cells (probably by endocytosis), and then rapidly translated into copies of the spike protein. In this manner, the adenovirus-mediated vaccines and the mRNA vaccines are doing essentially the same thing - providing a nucleic acid blueprint for making the spike protein. The adenovirus (a DNA virus) likely lasts a little longer (these are non-replicative adenoviruses that can't make more of themselves), and has a DNA blueprint for making the mRNA that will then get translated into the spike protein. So it's essentially doing the same thing as the mRNA vaccines, but is one step further back in the process.
Anyway, one of the limitations of gene therapy has been the inability to get the corrective gene sequences to last. This can be overcome with some viruses (HIV-derived vectors, retroviruses, AAV), but remains a big limitation. That said - it's not a worry that the Covid vaccine mRNA is going to hang out and make spike protein for a long time.
That said, the antibodies that are made could, theoretically, have some cross reactivity to human proteins and thus cause some unanticipated autoimmune phenomena. This is a risk with any vaccine.