And also, since the definition of a man is an organism that has sperm, and the definition of a woman is an organism that has eggs: I believe if only two "men", only two "women", or someone on their own can create an embryo/baby, then there is no difference between man and woman, egg and sperm. The only difference between egg and sperm was we thought only sperm fertilized egg, nothing else.

You've started out with a false premise based on erroneous definitions.

No, the definition of a man is NOT "an organism that has sperm," and the definition of a woman is NOT "an organism that has eggs."

A man is an adult human male; a woman is an adult human female.

Biologists base their definition of male and female on whether an organism during early development formed the reproductive anatomy that provides the potential capacity to produce/generate/release either sperm or ova - the male and female gametes, respectively - at some point later in life. Biologists use this gamete-focused definition to divide organisms that reproduce sexually into two distinct, mutually exclusive categories - male and female - because it's applicable to all sexually-reproducing species across the board - plants and animals alike.

During the earliest phases of life, all individual plants and animals belonging to sexually-reproducing species will develop along one of two different pathways - male or female - based on their DNA.

In humans, an individual on the male pathway will develop whilst still in utero reproductive anatomy organized around the potential capacity to produce sperm, aka male gametes, at a later point in life. Sperm are made in the testes, the male gonads.

A human individual on the female pathway will develop whilst still in utero reproductive anatomy organized around the potential capacity to mature and release eggs, aka female gametes, on a cyclical basis at a later point in life. Eggs reside in and later will be released from the ovaries, the female gonads.

Sex differentiation in humans begins very early on - the two different types of gonads start developing at 7 weeks, but myriad sex differences in embryonic stem cells have been found at 6-7 days and most likely exist from the start. By current methods of genetic testing, CVS and the NIPT, sex chromosomes and genetic anomalies, including those affecting sex development (DSDs) can be ascertained at 8 weeks. Visible sex characteristics can be easily observed by standard fetal sonograms that are customarily done at 18-20 weeks - and now are often done earlier for "sex reveal" reasons. (BTW, the normal gestation period for a human is 40 weeks.)

Male humans whose development proceeds typically in utero and afterwards will normally start to be able to produce sperm during puberty, which usually begins age 10-12. Male humans able to produce sperm usually will continue to do so the rest of their lives. When males make sperm, they do so in very large quantity - about 100 million sperm each time.

Female humans whose development proceeds typically in utero are born with all their eggs already in our ovaries. In the early stages of development, female humans will have about 6-7 million eggs that are extremely fragile and soon begin to die off. By the time she is born, a female human will have only 1 million eggs. By the time she reaches puberty, she will have circa 300,000 eggs left.

During puberty, female humans typically become able to mature and release our eggs, in a cyclical process that occurs monthly called ovulation.

But unlike male humans, who produce millions of sperm each time they ejaculate, female humans usually mature and release only one egg each time we ovulate - or in rare circumstances two or a few eggs. (Medical intervention with drugs to hyper-stimulate the ovaries is almost always required to get a woman to release more eggs.)

Whereas human males with normal sex development will have the capacity to produce sperm from puberty to death, human females with normal sex development will have the capacity to mature and release eggs only for a portion of life - from puberty until menopause. Since the average age of female human puberty is 11 and the average age of menopause is 51, this portion of a female human's life usually lasts about 40 years. Given that the average human female lifespan in many countries nowadays is 84+ - and some places it's nearly 90 - this means a majority of female humans who live a full lifespan will naturally have the capacity to mature and release eggs for less than half our lives.

Some humans will never have the capacity to produce, or to mature and release, gametes because they have very rare medical conditions that cause sex development in utero to be atypical. Some humans will develop the capacity to produce and release gametes as expected, but for various reasons their gametes won't function normally, so reproduction by natural means will be difficult or impossible.

Many other humans will lose their ability to produce, or to mature and release, gametes over the course of their lives due to accidents, disease or medical treatment such as chemotherapy - and because for female humans, losing this ability is the natural result of menopause.

In humans, being male or female is not based whether any of us "has" sperm or eggs right now this very second the way OP says. Nor is it based on whether we were able to produce or to mature and release gametes at every moment of every day in the past, or will have the capacity to do so every moment of every day in the future. The human capacity to generate/release ova or sperm naturally varies over each of our lifespans.

Human males can't make sperm before puberty. Human females are born with all our eggs, but we only obtain the ability to mature and release them at puberty - and we naturally lose this ability at/after menopause. Moreover, even during the years of our lives when female humans are capable of maturing and releasing eggs, we don't do so in vast numbers or spontaneously upon orgasm the way male humans do with sperm. We release our eggs one by one in a cyclical process that is not prompted by sexual arousal, orgasm or sexual behavior, and which we can't make happen by having sexy thoughts, watching porn, masturbating or by closing our eyes, making a wish and commanding our bodies to "ovulate now!"

Generally speaking, a human female will release one egg a month, or 12 a year. However, between menarche and menopause, most girls' and women's ovulation cycles will vary in length - so some years, an individual will release 8 or 9 eggs, other years twice as many.

TL, DR: The key element in how biologists define female and male in all sexually-reproducing species is which kind of gamete - egg or sperm - that early in development an organism developed the potential capacity to produce at some point later in life. Not whether one "has" or can generate sperm or eggs right now this very second.

As for all the rest of your post, none of it makes logical sense to me. Humans have figured out all sorts of ways to outsmart, "conquer" and attempt to override nature, but none of those efforts mean nature no longer exists.

Even if scientists are successful at bringing animal and human offspring into being without sexual reproduction, chances are that as sexually-reproducing plants and animal species will still continue reproducing the old-fashioned way that's been going on for the last 1-2 billion years.

The development of agriculture didn't stop animals and plants from growing in the wild. Plant hybridization and animal breeding haven't stopped evolution amongst plants and animals from going on naturally. Cloning sheep and making mouse embryos through artificial means in laboratories won't stop sheep and mice who live outside those labs from reproducing through sex. The invention and use of IVF, IUI and other methods of medically-assisted reproduction doesn't mean most of the human race has stopped conceiving babies by fucking - or that most people would prefer to do so, either.

Also, whether the offspring of sexually-reproducing species comes from the merging of ova and sperm or through some newfangled method that mad (and almost always male) scientists with god complexes come up with, the offspring themselves will all still have a sex - won't they?

NONE of the stories of the lab feats you seem so excited about say the cells that these scientists are working with - and all the embryos they are creating, or trying to bring into being - lack sex chromosomes. [Edit to add: links I've posted in other replies reveal that the cells these scientists are working with are typically male cells. Coz sex discrimination against female cells is the norm in lab research.)

Also, you might want to look into Lysenkoism.

Finally, enthusiasm for news reports that say such things as

Scientists say early experiments suggest it may one day be possible to make babies without using eggs.

“We’re talking about different ways of making embryos. Imagine that you could take skin cells and make embryos from them. This would have all kinds of utility.”

Artificial wombs and embryos made from skin cells – remarkable new techniques could revolutionise reproductive biology

And

A new paper published in Nature journal by a team led by professor Nicolas Rivron, from the MERLN Institute of Maastricht University, is making headlines around the world. The research sees scientists grow a very early stage embryo in a laboratory without eggs or sperm.

Seems highly correlated with misogyny. This can be the misogyny of boys and men with deep, abiding womb envy and rage and covetousness towards females - or it can be the internalized misogyny of girls and women who loathe their own bodies; look down on their own sex for having the capacity to conceive, gestate, give birth to and breastfeed babies; and often have serious issues with their mothers and others they perceive as mother figures.