So then how can it be such a thing? If it doesn’t actually exist.
By the way, what do you mean by a magnitude? A set of numbers?

A number is a point on the axis, but an infinitesimal is not a point but a segment of arbitrarily small length.
I didn’t say that the infinitesimal is the number closest to zero; I only drew your attention to the fact that there is no such number, and therefore there is no logical contradiction here.

That’s not a correct definition. If it is an existing number greater than 0 that is smaller than every number that exists, then that means it is also smaller than itself. (Or alternatively, it will always be greater than its own half.)

The correct definition is that it is smaller than any number you choose or can conceive of—not necessarily every number that exists.
For example, the reciprocal of the number TREE(3) is a number that exists, and it’s not clear that it’s a number one can actually conceive of. But since that was defined, this includes it too.

In calculus, if I remember correctly, the lecturer said that in the past an infinitesimal was conceived as a positive number smaller than any other number. If I remember correctly, that was Leibniz’s definition (his notation also gives that away). But afterward the name remained even though the conception changed.
In fact, it took a long time until infinitesimal calculus was properly defined, thanks to French mathematicians who did the grunt work of building a systematic theory.

What was hard for me to grasp was the definition of a real number presented as a decimal fraction, as the limit of a sequence (and that’s also why 0.99999… equals 1).