Left- or right handed or something else?

Let's start with the things we don't know:
We have no idea how many left handed people there would "naturally" be in a population as no scientist is able to make a claim to have discovered a population that is not under some cultural influence on handedness with differing and evolving numbers.
Current handedness "tests" produce the statistics of the culturally influenced numbers of left- and right-handers. Often this numbers are equated with something biological for further research.

There's one profession that is knowledgeable about getting the use of left and right hand in accordance with disposition. - drummers
Drummers know a lot about using the correct lead hand when drumming, because violating that rule gives everyone, except the lucky few true ambidexters, a considerable headache and mental exhaustion.
The dominant drumming hand is not necessarily the most comfortable writing hand.
There are drummers who feel comfortable using different lead hands for different kinds of beating techniques like forehand and backhand beats or with big instead of small drumsticks.
Laterally cross-dominant is the term used for this configuration. A lot of people know that they are to some degree cross dominant because they know a specific ability that is better developed on their non-dominant arm or hand. Often they find out just after having the other one temporarily immobilized. Cross-dominance has not yet been understood as there are people around who jump for height with one leg and for width with another.
Baseball is one of the few sports with attention to cross-dominance that provides advantages.

A number of people commonly list this incomplete list of specific lateraly superior skills for their arm or hand: Testing can be a lot about internal perception of better capabilities.

strength
endurance
throwing (speedy muscle contraction)
3-dimensional control
finger motoric control

Other than the hand or arm, everything has lateral dominance with no correlation being established between the lateralization of one kind of organ and another.
There's a simple test on some kinds of ocular dominance:
Point with your finger at a target with both eyes open. Afterwards you close each eye in succession and check for which eye the finger is closer aligned to the target. Next step is checking with targets at different distances.

It's normal for all children to have a phase when they try to figure out their constitution of specific skill lateralization. A long phase of such testing is often referred to as "insecure handedness" (the organ under most cultural pressure for conformity). It is possible to solve the same task with recourse to different kinds of skills that can be developed on one side or another. Cooperation between attributes on the same side is usually quicker to learn. It is not known to what degree concepts like being left- or right handed can influence a person on understanding themselves and reduce the employment of attributes situated on the non-namesake laterality. Generally, writing from left to right looks better when done with the right hand, but it helps to learn a few tricks from Arabic calligraphy that has a lot of right handed artists writing from right to left. There are two different writing styles of left-handers, over- and underliner. Both develop natural with rather different calligraphic looks. They can depend on different unknown combinations of opportune skill sets.
Knowing a configuration of skill lateralization does not determine the laterality employed because well-known skills can often be used to solve the same task under different mental approaches. For this reason cross-laterality is often listed among ambidexterity with consequent scrutiny because obvious differences in capability could be detected.That did not fit the definition of ambidexterity as developed from ancient sources.
Ambidexterity was one of the educational goals of Renaissance men. The enforced approach of teaching absolute skill equality of both hands irrespective of any disposition seems to have been retarding for mental development of children.

"I was left handed and was forced to become right handed" is a frequently reappearing topic throughout the world. Often it will result in surprising difficulties to use the old envisioned natural hand. In order to not use a hand, that would be preferred because of an inherent skill set initially considered suitable, you need a mental construct that helps to learn a different approach with corresponding other existing skills. The task has mentally been connected with a specific skill set approach and the formerly more convenient approach of skills is not activated. With the capability to figure out and activate specific skills flexibility in approach can be obtained.

An inherent skill set considered suitable should be remembered when a child starts writing as there might be a misunderstanding about this new kind of task.