There is a question on stackexchange that is not only related
Why cytotoxic T cells don't kill dendritic cells when they present antigen?
Your question refers to "some cells" (explanatory text) and "transfected cells". Thus, any cell that takes up the m-RNA-vaccine is referred to. However, any answer must be based on the different types of cells involved in immunization, vaccination and - in killing infected cells, in the end. There are the immune cells which present antigen, there are, as well, the body cells which present.
The explanatory text of your question speaks of "presentation". The very first paragraph of Wikipedia on "Antigen presentation https://en.wikipedia.org/wiki/Antigen_presentation seems to even limit the term "presentation" to presentation by immune cells, which would exclude any body cells, like muscle cells transfected with vaccine antigen. This might illustrate that immune cells are different from body cells in respect of presentation of antigen that becomes target of cytotoxic t-cells.
Both immune and body cells take up antigene, however the immune cells, by presenting prime naive T-cells, i.e. activate the T-cells. The related question cited asks about the fate of the immune cells.
As for the fate of body cells like muscle cells you refer to in the last paragraph of your question the video message you link to in your own answer seems to refer to those body cells: transfected cells will simply be killed, but not enough of them to cause harm.
Here are additional remarks, enjoy: I think it is an open question if the transfection as such would lead to cell death (assuming some failure of cytotoxic T-cells, not doing their job). Think of vaccination that transfects with "attenuated", if not live virus: even if the virus is incapable of - natural - replication it is unlikely that the host cell whose ribosome machinery has been usurped can keep up its metabolism. I would like to be allowed to mark the following as a new thought of mine: the fact that the antigen is being presented speaks in favour of the cell not being able to survive - which is counterintuitively (how can a cell infected by virus still be able to digest, process, transport and present antigen on its MHCs, at all? Looks like it will stay alive.)
From the very fact of antigen presentation it is possible to infere, in my opinion, that the cell will not survive transfection. Host cells may have their own, inner defence mechanisms which are able to beat infection without any presentation of antigen - think of "interferon", but also, nowadays, of double stranded RNA, dsRNA. A cell that is successful in defeating the virus without engaging the adaptive immune response would cancel out its survival by presenting and attracting cytotoxic t-cells. So it is a possibility that cells in spite of being transfected by mRnA vaccine do not die. However, that would be considered a vaccine failure.
So with effective vaccination transfected body cells will die.
Caveat: from my reading of textbooks it is a possibility that vaccination might be effective even if body cells are not transfected, but that is beyond your question. Premise of your question is that a cell is being transfected (then what happens?).
A reference on the possibility that a "transfected cell" might survive by self defence would be "RNA interference",
Last not least - see the video you linked to - there is "B cell immunity".
To my mind it is a possibility that B-cells become transfected by mRNA vaccine - if so, they should present the antigen as an endogenous on to cytotoxic T-cells which makes your question very relevant in respect to B-cells, in my opinion. However, B-cells produce antibodies, considered most important in vaccination. This speaks in favour of B-cells taking up antigen via their B-Cell receptor that should be among cell debris of transfected cells that died - for instance few muscle cells transfected by vaccine. What you need here, in respect of B-cells is authority on either cross-presentation of endogenous antigen by B-cells (if mRNA is being uptaken through membrane, if B cells are transfected) or, authority on B-cells "only cleaning" up dead cell's antigen as part of dead cells, killed by virus or immune cells.
My answer is in line with the answer you already gave - elaborating on what is in fact causing the cell's death:
"Do mRNA vaccines cause transfected cells to be killed by cytotoxic T cells?"
Yes, mRNA vaccines cause transfected cells to die.