You are talking about Alternation of Generations - gametophyte and sporophyte. And the production of gametes is NOT asexual. So the answer is: yes. Meiosis always "scrambles" DNA in chromosome pairs when crossing over occurs. It is sometimes called recombination. In evolutionary terms it is known as 'meiotic drive'.
Somewhere you got the word 'spore' misplaced in your understanding. Flowering plants produce meiotically generated spores (gametes): pollen grains, for example. Not part of asexual reproduction.
Cell division or mitosis is the driving force for asexual reproduction. Gametophytes like molds can make spores by mitosis, even though they are haploid.
All of this gets messy and confusing to students when you look at fungi, because most times, the organisms we can see are gametophytes - they have half the normal number of chromosomes.
Anything with one n chromosomes is a gamete in the sense of gametophyte. Production of more cells by mitosis does not change that.
Anything with two n chromosomes is not a gamete in the sense of gametophyte versus sporophyte.
I see what your drawing shows, and yes they use the word "spore". IMO, bad choice. Why? Because you are confused by it. Spore has multiple meanings. It usually means asexual reproduction. You are correct in that usage. Why? If your instructor thinks that is the correct way to go you are stuck with it.
If you talk to a Botanist they will explain it a little differently.