No it isn't necessary for you to say any specific words or have a particular intention for purifying something; you may just do the action. Yes other acts such as wudu, ghusl, salah and fasting etc would require an intention.
Istinjā` is washing yourself after relieving yourself in the toilet. This is of course very important for the purpose of tahārah, wanting to pray, etc.
However, there is something else that is important to do, but is not obligatory, and that is istibrā`. Istibrā` is clearing the urinary outlet after urinating. For a male, you do this by what we call the nine squeezes.
The advantage of istibrā` is if you do it, with any discharge you might feel after the istibrā` and the istinjā` , you consider it to be tahir. However, if you do not do the istibrā`, and you experience a discharge, not knowing what it is, you deem it as najis, and you must wash yourself again.
I have added a link to a recorded lesson on this, with detailed explanation.
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