No. Please see the link below for the 45 international trademark classes - the services are at the end of the list. This schedule pre-dated the internet, so "online" isn't even mentioned, although computer things are mentioned.
Applying for a trademark registration is not as easy as it may look, and about 1/2 of all applications get rejected, so don't waste your time, effort and the nonrefundable fees --hire a qualified IP lawyer.
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No and to be candid, your question shows that you should not be handling this on your own. You need a trademark lawyer. Otherwise you are going to waste a lot of money on a trademark application that may prove worthless. Selecting classes for your trademark is a rather complex matter that takes experience and judgment, and you are simply not in a position to do this by yourself. More generally, applying for and prosecuting trademark applications is not something that lay persons can handle well. If you don't have a budget for trademark counsel then, in all candor, you probably are not ready to file for a trademark.
No they are not. Web-based service are most often grouped under the category of the services to which they relate rather than whether they are online or not. For example, an online tire store would compete with brick-and-mortar tire stores and would be grouped under the service category for tire stores. Similarly, an online dating service would compete with telephone dating services or brick-and-mortar dating service offices, and all would be categorized in the same class.
That said, however, the reputation of a well-known online site might be sufficiently great that its coverage would extend to other classes. For example, if you were to set up AMAZON job posting services, you can be sure you would hear from the trademark attorneys for Amazon.com with a demand that you cease and desist from your infringing activities.
Trademark law is superficially simple but in reality very complex. You need a trademark attorney to help you find the legal pathway through the legal minefield.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
Web based services may fall into the same class, but the distinction and whether or not you will get a trademark registered depends on the area of business and your description on the application. The Examining Attorney looks at several factors in evaluating new applications against pending ones, and the analysis is not always straightforward. If the registered mark is already considered famous, then there is even less chance of getting your mark through. In the end, classifications are not determinative but rather the similarity of the marks and goods/services is controlling.
This is not a legal opinion, and should not be relied upon as such.