Does it prevent the trademark, such as the name and logo of the project from being used or modified
The limitation does not prevent this - but this is because it is the default in law for any names, logos etc which meet the threshold to count as a trademark; it it likely that the names and logos of most open source projects do not meet that threshold. What choosealicense is doing here is classifying licenses into two sets:
- Those which make this explicit.
- Those which do not make this explicit and instead rely on the default in law.
Realistically, the difference here is probably moot, but if it's something that you have a specific concern on for a specific project, you should take professional advice.
or does it stop contributors from registering their own trademarks regarding the name and logo of the project?
It is very unlikely that contributors would be able to make a successful trademark claim for the name and/or logo of an open source project. They are, by their very nature, something available "in the commons" and not indicative of a particular relationship. So again, this is not prevented by the limitation but instead by the very nature of trademark law.
or in other words, to patent the project
Patents and trademarks are very different things. Do not confuse the two.