the bank’s internet(Internet) service

according to my spell checker, always.

The passive, Jinmin.

And [what time] when should "internet"____ initially capitalized?

I agree with this opinion:

The argument for capitalisation is simple: there is only one Internet, just like there is only one Moscow. Sure, it's made up of millions of elements but so is the Great Barrier Reef. Therefore Internet is a proper noun and must have a capital. Until quite recently, this opinion held, partly because of its inarguable correctness. But good English isn't necessarily about correctness " usage plays an important role.

To people who have grown up with the internet (and let's face it, that's anyone born after about 1985), the internet is as ubiquitous as the phone and the radio, and probably doesn't seem worthy of special treatment. The word is showing signs of "maturing".

The irony is that the internet itself (especially email) has been partly responsible for the overuse of lower-case lettering in almost all design contexts. People who capitalise their initials in their email address just look weird!

Therefore Internet is a proper noun and must have a capital. ... To people who have grown up with the internet ... the internet is as ubiquitous ... The irony is that the internet itself ...

?????

Either one gets the point across. Usage is kinda going both ways right now.

Same thing with E-mail. Traditional rules indicate that "E-mail" should be capitalized, the same way one does with "A-frame" or "T-shirt", but actual usage is leaning toward "email".

What I'm seeing is a tendency to overcapitalize all abbreviations.

Well, as names are capitalized, Email should be capitalized: