Thanx for the update, Chandan. It appears I was right, Twitter is not super-protective of its name. But its policy of being lenient raises other questions, which may require a lawyer to answer.
It appears Twitter wants to take the position that it will allow some use of its name by third parties, but try not to allow some other use of its name by other third parties. I am not sure whether courts allow this selective use. My understanding is that courts like all-or-none decisions. If Twitter sues all third party users, courts accept that. But courts don't like it when Twitter decides it will sue here but not there.
Think of this analogous situation: people (in the US, at least) have privilege over their medical records. Third parties do not have access to a person's medical records.
Let's say husband and wife sue each other for divorce. Their medical records cannot be used as part of the divorce proceedings. But let's say the wife, for example, says husband drove her crazy and she started seeing a therapist. The moment she write words like that in her legal papers she waives privilege to her medical records and her husband has free access to them. Her privilege is gone.
In the same way, if Twitter permits some use of its name, but tries to stop other use of its name, I am not sure whether courts will allow selective use at Twitter's discretion. It is possible (lawyer here please) courts will say either you disallow everyone or you allow everyone, you cannot pick and choose.