Is this sentence correct? Any logical or grammar problems?

1. Change choose to chosen or change have choose to chose - this is past tense as you've already selected the group name. Either form I've suggested is acceptable.
   
2. less directed what? The clause makes no sense without further information
   
3. Your second less should be fewer - it's fewer for enumerated things, less for collective stuff like liquids - e. g. less coffee, fewer coffee beans
   
4. Change meant to means - now you're in present tense; you are describing what people currently know, in the present time
   
5. Don't capitalize Therefore, or change the semicolon that precedes it to a period. I suggest changing the semicolon to a period.
   
6. You may wish to capitalize your group's name
   
7. Change related to relate - same as meant and means - you need to be in the present tense as you are describing what people currently relate to, right now
   
8. The whole thing as it is reads incredibly pretentious.

Introspection as a word choice is a less than optimal choice. It’s not a grammatical issue, however. Introspection implies a person looking into one’s soul as opposed to examining or studying of words. Perhaps using something like Word Study Group, Word Sleuths or Word Detectives.

You’re correct in not choosing rumination.

Style wise, you need to study how to keep your tenses consistent and in the present tense.

Use the phrase fewer people instead of less people.

Furthermore, I’d be remiss in not correcting your usage of the phrase ‘have choose’ instead of ‘have chosen’.

We have chosen the name “Introspection” for our group.

Just about everything's been covered about your sentences. I'll only add that your thread title should read "Are these sentences correct? Any logical or grammatical problems?"