A more common version of "I take your point" is "I see your point". This means the person understands your point of view. Also there is another shorter version which is "point taken". Yes, understanding and agreeing with someone's point of view are two different things. However, if the person agrees with your opinion then most likely they'll say I agree with you. If they say I see your point, that doesn't necessarily mean they agree with you. Hope this helps.

"I take your point" is not a native English phrase in the US. Agreeing and understanding are two different things. Agreeing phrases commonly used in the US: "I agree." "Agreed." "I agree with you." "You and I are on the same sheet of music." "Exactly." "I hear you." Understanding phrases commonly used in the US: "Understood." "I understand." "I understand your point." "I hear what you are saying." "Makes sense". None of these phrases indicate you agree - only that you understand. Accepting someone's point of view means that you accept that they have their own point of view - whether it is similar to yours or not - and that you aren't angry with them about it. You are ok that they have their point of view. Hope that was helpful. :)

I understand it to mean “I understand your argument”, but according to Wiktionary ( https://en.wiktionary.org/wiki/take_the_point#English ) it can also mean to agree with your point of view. I suggest you ask if in doubt. I think people will quite often say “I take your point, BUT I think that...” If they just say “I take you point” and nothing else, you’d better ask if that means they agree.