Is it ethical to ask my students to sign up to this Web service
through my referral link, given that I use this Web service in both a
personal and professional capacity under the same account? Let's say
that I tell them everything that I mention in this post, too
(including my usage of the service).
Before answering the "is it ethical" question, let's recognize that this would create (at least) the appearance of a conflict of interest: by managing the course in a way that benefits you monetarily on top of your normal salary, you can cast doubts in the eyes of the students or other outside observers as to whether your actions are purely motivated by the desire to get the best educational outcome or are influenced by the external, and potentially conflicting, interest of making money. Now, it is implicit in your question that you recognize this issue and are trying to think of ways to allay those doubts. Unfortunately this is not as simple as it seems. Consider some of the things you are proposing to do:
Advertise publicly that you believe in this product and use it yourself.
The problem with this is that those actions are also consistent with the actions of a person who is "in it for the money" and doesn't actually believe in the product, but wants people to think that he does.
State emphatically that the students' use of the service is voluntary and absolutely not required.
The problem is that this could still exert a subtle pressure on the students and signal to them that you would be happier with them if they used the service. You are an authority figure and hold substantial power over the students, and some of them may want to curry favor with you, either consciously or subconsciously, and might therefore use the service even if they don't think it's useful to them. Also, even if only a few students end up being influenced to use the service by your recommendation, you are still benefiting monetarily so this does not remove the suspicion of an action motivated by a conflicting interest.
Share the link to this Academia StackExchange question with your students, to let them know you have thought seriously about the ethical implications and strengthen their belief in the nobility of your intentions.
Again, the problem is that an insincere person whose intentions are not noble might behave in exactly the same way.
To summarize this part of the answer, although it seems likely that your motivations are pure, which I assume is why a wise person like Pete L. Clark decreed in his answer that there is no conflict of interest here, it is still the case that there would be an appearance of a conflict (which would also mean that at least in principle there could be an actual conflict), and it is not obvious how this appearance can be completely removed.
Now let's turn to the ethics question. I happen to be on a committee of my institution that oversees potential conflicts of interests (which we refer to as COIs). The context is different (COIs that arise in scientific research funded by a mixture of public and private money on topics with commercial potential like pharmaceuticals) but many of the issues are similar to those raised by your question. One of the first things you learn when working on such a committee is that academia has many areas with a strong potential for conflicts of interests. Unfortunately it is completely impractical to take an approach that would simply forbid such conflicts to exist (e.g., by forbidding research to be funded by an entity with a commercial interest in the outcome of the research), since that would mean that a lot of very important research would simply not get done. So, the question becomes instead how to manage the conflict by taking steps such as disclosing the conflict in various ways and other measures.
The point I am trying to make is that just because there is a potential for a conflict doesn't mean that the proposed action is unethical. However, it certainly means that special care is necessary to make sure you are not even perceived as acting unethically, and it also means that there would have to be a fairly compelling reason for the proposed action to be taken. In your case, I have to admit that I am not seeing such a compelling reason. The small monetary gain and other benefits you are likely to receive from the referral links are overwhelmingly smaller and less significant than the benefit of a research that could lead to the development of a new drug or medical device. So I think the risk that your actions would be perceived negatively by your students or employer in this case far outweighs what you stand to gain. My recommendation is therefore: don't do it. The only exceptions I would make are 1. if you ask your department chair and he or she specifically approves this; or 2. if you announce that you will donate the proceeds you will make from the referral links to charity (and specifically, a charity whose mission is completely uncontroversial and could not possibly be frowned upon by any of your students).