Very good. I like you to correct my understanding:
My limited understanding on speciation is something like a population of one species is isolated into two populations. So the two groups stopped interbreeding and developed independently on their own traits. This is not a strict definition, but is a description. Biologist could phrase this idea by a more precise language.
Yes, that's roughly the
biological species concept I mentioned: a set of organisms capable of interbreeding. Speciation in this case means reproductive isolation. It's quite obvious why this doesn't apply to
any asexual creature (not just bacteria: even some lizards reproduce without sex).
So, bacteria split to populate. I am not sure how does algae or fungi populate themselves.
Sorry, I don't understand this bit
But until they evolved into something which will "give birth" (bisexual?) to their offsprings, there is no speciation to me.
A correction: "giving birth" and sexual reproduction don't necessarily go together. Many single-celled eukaryotes have proper sex (as in undergo
meiosis and fuse two sex cells). I wouldn't call that "giving birth" but it's definitely sex.
Any life populated itself before the bisexual function kicked in, is grouped and labelled by me as a "life not evolved". I know there are/were a lot different life forms fall into this category and need some systems to subdivide.
And that is why your lumping of all bacteria into one giant "species" is unfortunate. It makes it sound like they were somehow less diverse or disparate than eukaryotes, which is very definitely not true.
But that is a problem left to biologist, not to me. So if you don't like this definition, you are extremely welcome to replace it with another term of a similar meaning.
I don't feel the need to replace the biological species. I'm not sure we should even use the word species where the BSD isn't applicable. But then I'm not technically a biologist (
yet 
)
So, to me, bacteria are all in one species, which is the (non-sexual + single cell) species. I will be convinced in this thread that bacteria evolved if someone showed me some bacteria that is populated by a sort of sexual function, I guess, in that case, it must have been a multicellular creature.
As I said earlier, you don't have to be multicellular to have sex (unless you have your own definition of sex, too, but I prefer to stick with the official one).
You may want to consider that bacteria do
exchange genetic information (and lots of it), although they don't do it in the strictly regulated eukaryotic way. They also do it with very distantly related "mates" - horizontal gene transfer even occurs between bacteria and archaea, which are about as far as it gets on the tree of life.
So, Naraoia, to your question, my thought is that the word species is not a proper word for the classification of bacteria. A different word should be used.
In that we agree. I still maintain that your conclusions that bacteria somehow don't evolve are simply wrong, though.
My scheme on this biological issue is very primitive and simple.
No problem with that, so long as you are willing to learn from people who know more about it.
But unless I am corrected on the fundamental level of definition and process, my logic on this issue stands. I understand miscommunication happened due to the different content of definition. But until this post, nobody is asking me what is my definition of anything (big credit to Naraoia, excellent student).
Didn't you have a whole discussion about the definition of evolution?
The only message I read is that my definition is wrong. If you do not know what it is, how could you say it is wrong?
Well, as for evolution, you clearly rejected the official definition. Of course "just change" is not the only way you can define evolution, but it has become accepted like that. And one of the good things about science is that you use terms with known, agreed-upon definitions so others know what you are talking about. If you mean a different thing then use a different word. If everyone just started arbitrarily redefining scientific terms chaos would quickly ensue.