Isn't a soloist someone who performs solo (not a solo)?

A soloist, as the dictionary tells you, is someone who performs a solo. Just as a novelist is someone who writes novels, not someone who is novel. Cambridge explains that the suffix -ist primarily denotes "one that performs a (specified) action".

The most common term for someone who performs solo is a solo artist.

I did a quick bit of research and there are a few, rare examples of 'solo' musicians being called soloists, but I would consider them to be outliers. For example, Smithsonian Music referred to Louis Armstrong as a soloist, and while he was known as a 'solo artist' who sang and played trumpet, he never or rarely performed without a band, and his trumpet playing was usually "a solo". If this is an acceptable second usage of the word, it seems rare, and debatable as to wether it still doesn't retain the dictionary definition.

A soloist is a) an instrumentalist or vocalist who performs alone unaccompanied by others, in their professional capacity as "solo artist" or b) an instrumentalist or vocalist who performs a passage of a piece alone while in the company of other instrumentalists or vocalists who have been performing as a group prior to the solo passage and who will recommence performing as a group when the solo passage is finished.

The performance itself of a passage alone as a sole instrumentalist or vocalist is often called "a solo" or "the solo", a shortened form of "solo performance".

There's a bit of history involved. For some musical practices and traditions, a single person with an instrument is the norm, like a griot with a kora or Homer with a lyre. But others have traditions involving ensembles. For these, a "solo" part is "alone" (Italian literal meaning of the word) in the sense that there is only one person performing that musical part, even if others are performing others at the same time.

Thus a choir of many singers can also feature a vocal soloist. There may be five or fifty sopranos all singing the choral soprano part, but the soprano soloist is the only person singing their musical part. Similarly there may be a cello concerto with a symphony orchestra; there may be a dozen cellists in the orchestra, all playing the same orchestral cello part, but the solo cellist is the only one playing their special part.

This usage was then borrowed by settings that are normally "one-on-a-part" anyway. For instance, a jazz trio of piano, drums, and bass might play together, then the bassist might "take a solo," a period in which their instrument is featured more prominently and is the focus of attention, more so than the other instruments. The piano and drums might keep playing during this solo, but less prominently. This usage might be the most counter-intuitive, since the bassist is the only person playing the bass part throughout the whole show, and is no more "alone" during their solo than at other times, but the term has been borrowed from other large-ensemble examples like the choir or symphony. (In fact, in a jazz "big band," there could well be multiples of an instrument. Like, there might be four trumpets who all play the same part together, but then one trumpeter "takes a solo" and is playing different musical material (most likely the other three take a rest!).)

perform solo and perform a solo are two different things though they are related.

A soloist performs solo OR performs a solo.

a solo performance means the musician or singer alone.

perform solo means without other musicians.

perform a solo means perform on one's own but with musicians waiting to play in an orchestra after the solo finishes.

So, they both mean the musician or singer performs alone, but the second means with other musicians or an orchestra waiting to play.

There are many different kinds of solos:

• piano solo, violin solo, etc.

Also, the OP posted this:

soloist(noun): a musician or singer who performs a solo. "he appears as a concerto soloist with all the great British orchestras"

and was surprised by the indefinite article.

Right, let's compare a and the here:

a musician or singer who performs the solo etc. would mean that the overall musical work has a solo piece and the musician performed that specifically.

The indefinite article a is used to express a generality. That dictionary defintion is a generality, thus, the article is right.