I thought the 'rowter' pronunciation re. computing was a purely American influence.
In the UK at least, router (in a computing context) _does_ rhyme with hooter. It is, after all, something that routes traffic -- exactly like your first example.
A router pronounced 'rowter', on the other hand, is a device for cutting grooves in wood.
Americans rhyme "route" (path) with "doubt" and "row" (paddle a boat) with "dough". If you are rhyming "row" (fight) with "cow", that's a word most Americans don't know.
The usual distinction is "to route" rhymes with "to doubt" and the word "route" is pronounced like "root". Some people also use the verb pronunciation for the noun as well.
Router as in the hardware always rhymes with doubter. I have never heard any other pronunciation for it. Route as a verb in IT always rhymes with doubt.
In the UK at least, router (in a computing context) _does_ rhyme with hooter. It is, after all, something that routes traffic -- exactly like your first example.
A router pronounced 'rowter', on the other hand, is a device for cutting grooves in wood.