In high school my teacher told us : $$f'(a)= \lim_{h\to 0} \frac{f(a+h)-f(a)}{h} \tag{1}$$
For a long time I've been thinking that it was an alternative version of the other definition ($\lim_{x\to a} \frac{f(x)-f(a)}{x-a}$) but now that I came across the definition of the derivative in non-standard analysis I realize her definition is quite close to the standard part definition $f'(a)=st( \frac{f(a+\Delta a)-f(a)}{\Delta a})$
Is $(1)$ correct or is it a bad-interpreted version of the "standard part" definition, from non-standard analysis ?
It is in fact just an alternative form. We have $$f'(a) = \lim_{h \to 0}\frac{f(a+h) - f(a)}{h} = \lim_{x \to a}\frac{f(x) - f(a)}{x -a} .$$ Simply set $h = x-a$; then $x \to a$ is equivalent to $h \to 0$.
In non-standard analysis one does not use a limit to define the derivative of a function $f$ at a point $a$, but uses the quotient $$f'(a) = st\left(\frac{f(a+da) - f(a)}{da}\right) $$ where $da$ is a non-zero infinitesimal element of $\mathbb R^* \supset \mathbb R$, i.e. a non-zero element of $$\iota =\{ x \in \mathbb R^* \mid \lvert x \rvert < 1/n \text{ for all } n \in \mathbb N \} .$$
Of course we can alternatively define $$f'(a) = st\left(\frac{f(x) - f(a)}{x -a}\right) $$ where $x \approx a$, $x \ne a$. Recall that $x \approx a$ iff $x -a \in \iota$.