Does meosis takes place in cell cycle?
This is a question of convention really.

When people discuss the cell cycle, they usually refer to diploid cells (for diploid animals). So in your context, M stands for mitotic phase and does not directly apply to meiosis.

In meiosis, you could say that the cell also undergo a M-phase (actually two in a row, see footnote) but then the cell cycle becomes irrelevant as you get 4 haploid cells (gametes) that cannot duplicate by themselves anymore, effectively breaking the cycle.

Footnote

The M-phase in the cell cycle is a simplification of the process of cell division as this phase can be further sub-divided into a Prophase, Metaphase, Anaphase, Telophase (PMAT for the purists) followed by cytokinesis. In meiosis, you have a slightly different PMAT process during meiosis I (i.e. 1 cell -> 2 cells) followed by another PMAT (2 cells -> 4 cells). You can find additional information here.