This raises an interesting question: where is the line drawn between 'freemium' and 'free trial of a non-free service'? Your mention of prompting after a set number of logins suggests that you're offering the free tier as a taster, rather than as a serious service.
It's a good point. Here is why we limit logins. We stumbled upon this actually and the instituting of login limits has resulted in incremental revenues.
One of the features of our service is a daily, real-time feed of company financings and M&A transactions. Before we had login limits, people would login every day and look at the feed but click on nothing so it would not trigger any 'usage' as we defined it. And they'd take the info presented in the feed and then go to Google and find out about the company or deal.
We were adding value to their lives else they wouldn't check daily but deriving no value ourselves. So at this point, we instituted the login limit so we could be compensated for the value we are delivering. Fortunately, many of those users have converted and are now paying us.
Of course, not everyone needs us daily so for those folks looking up a handful of company or VC or acquirer profiles, our free offering is plenty for them.
As to whether we are freemium or free trial, not sure. Think we take from both schools of thought but ultimately, irrespective of what we call it, the goal remains the same - build a product people derive enough value from that they're willing to pay for that value.
According to most of the freemium gurus, freemium and free trial are not the same thing; the former is defined as having a level of service that is free forever. Obviously free trials are by definition not free forever.
Freemium is a new word. A similar concept is 'crippleware' - you can do everything but not print. Or you can print, but it'll have a watermark. Or you can only save at certain resolutions. Etc.
I guess they would have been called freemium if the word existed then.