Where I work, if college kids do well in their internships they get offers before they even graduate.

The University of Waterloo advertises it as a feature of their co-op program.

Four month work terms alternate with four month school terms. Do six of them before you graduate and you have two years of work experience.

I did an internship every summer during college, but only my last 2 internships were software related. A couple of internship barely qualifies you. In retrospect, I should have asked my intern employers to extend the internship part time into the school year. Then, I think i'd be qualified for entry level. I got lucky on my last internship, tho and they hired me. At most I would count 4 summer internships as 1 year of experience.

You should definitely count any part-time tech work or summer internships towards total industry experience. Companies may not count it at first, but it will end up giving you leg up. It may not seem like much, but compared to the people who got a degree without working in the industry, I guarantee you know way more and would be more useful to a company.

In my case, I got hired at an "entry" level position for a huge company along with other "new college grads" types, many with masters degrees but no experience. I was promoted within a year to the mid-tier position while my less experienced cohorts spent 2-3 years getting there. From mid-tier, it becomes more about what you actually know and what you've done rather than "years since graduated from college", so getting to senior level with only 3-4 years "full time" experience is not impossible if you play your cards right (and really do get good experience during college).

It's not like you have to pay to submit the application/resume, so you could just try it and see, no??

Absolutely, assuming you had some responsibility and worked on an interesting project or two during those summers.

none of it really counts. What really counts is your skills and ability to confidently talk about them. Or put differently, its your choice if internship or whatever experience really counts. You also have to convince the prospective employer to respect your choice, but if you can confidently represent yourself that may be easier than you think.

I did one internship.

Traditional universities have "summer vacation". :)

Not at the vast majority of schools in the US.

Depends on the country I guess.

In New Zealand it's mandatory for Software Engineering (and other engineering), but not for Computer Science, as it's a science degree.

It's not mandatory. But it's incredibly hard to get hired out of school, even a top one, if you don't do any internships. You gotta grind from day 1 in this world.