What exactly is current?

The idea of current has been presented to me so many different ways I thought id try and find out a little more into what exactly is current.

Most places refer to a flow of charge as a current, seems good to me, but - I heard the current may flow outside the wire in... I want to say charge desnty fields but don't quote me on that... Is this is case? if so how?

Secondly the concept of "current flowing" seems to me very strage - if current = $\frac{\delta q}{\delta t}$ surely this implies the "current flow" people speak of is : $\frac{\delta^2 q}{\delta^2 t}$ ?

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 I cant answer the first question, but as to current flow verses current, the former is just bad grammer. it can be useful to specify the direction which the current is in. this probably comes from the usual way people are introduced to current, as one of three objects in Ohm / Kirchoff Laws. often, an understanding that electrical current is a flow of electrons comes after the basic formulae, and after we have learned to talk about it.
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What exactly is current?

I will defiately be more careful as to "current flow" in the future in terms of the model shown I assume this is the exact same for positive charge carriers? Not in a metal but I remember reading they do exist (Ice perhaps it was?).

Do we perhaps talk about current too often when we really mean to speak of the charge?