You asked me if it's wrong for a girl to make the first move, and if it's not, why then we have the peribahasa "perigi cari timba".
My answer - no, it's not wrong for a girl to make the first move. And the 'perigi cari timba' is a cultural thing, not Islamic, so I guess we are not strictly bound by that.
Having said that, it does not mean that one could sesuka hati make the first move. Saidatina Khadijah made the first move in proposing marriage to Rasulullah, but it was done discreetly, in a way that would not dishonor any party had the proposal been turned down.
The main point is - it must be done in a way that is discreet and honorable.
Meaning - the girl might want to try using a trusted middle person.
Preferably someone more mature, married and trustworthy.
And I'm talking about making the first move towards matrimony - not for "nak kawan-kawan" or "nak kenal-kenal" without any plan for marriage.
If it's just for "saja suka-suka", then you'd better go ask others in dating sites.
I guess many of us still believe that the burden of expressing an interest lies heavily on the guy's side rather than the girls. Yup, the good old-fashioned way. In fact many girls are afraid of being labeled "tak tahu malu" if they were to make the first move, like it's a huge turn off or something. This is where the importance of discreet execution comes in...
"Making a move is like submitting a tender", or so my former boss, Encik Dollah claimed.
"Before one submits a tender, the company would study 'the project' thouroughly - to know the cost, the expectation, preparing the budget and all.If 'the project' is beyond the company's capability, then it would not submit any tender at all.
However if the tenderee is actually interested in the tenderer, then the tenderee could invite the tenderer for direct negotiation, and they might be able to work things out after all."
Similarly, after studying a particular girl he might be interested in, a guy might not make the first move if he thinks that this girl is out of his league. However, should the girl make it known that she actually has interest in him, that the interest is mutual and that she is not actually out of his league, things might be different.
"If the girl is brave enough to make the first move, we don't think of her as being bold or agressive. I believe the guy would be grateful for her to ease some pressure off his shoulder. Who said that guys are not afraid of rejection? Sometimes guys are more affected by rejection compared to girls. Seriously"
So there C, my answer to your question.
Personally, I don't think there's anything wrong with a girl making the first move.
Only that it must be done in a way that would not cause any party to lose face.
After all, shyness is part of faith.
It's right to be shy, but it's also not wrong to take the first step.
If you have performed istikharah and the sign is positive, then you go girl!
If your nawaitu is right, insya Allah, Allah will guide you towards Him.
Have faith and be strong.