Especially since you are new, there would be nothing wrong with ringing round everyone, asking them to confirm that you have accurately captured their thoughts. Or send out your first draft as a draft copy, asking everyone to confirm by that you have accurately captured the actions. And/or could you pick one or two people to review the minutes before you issue them, to check that you have accurately captured the sense of the meeting? Its in everybody's interest to make sure they are accurate.
You could also, when you have what might be an action, repeat it back to the meeting to ask that you have captured it accurately and at same time, ask them to confirm by when and by whom it will be completed. I do this, i find it really helps the meeting to all share the same understand of what will happen next. It's not all that unusual for the discussion to break out again, and an entirely different action to be finally agreed. You are adding value to the meeting, not wasting their time.
Even with these reviews, I would always include a sentence at the end of the notes, asking anyone who remembers anything differently, to contact me in the first instance. Noone ever has, but it puts the onus on them to update you, and removes any excuse to say 'but i said.... Not ....'
It's all about putting the onus on the people attending the meeting to make sure they have been clear.
All the best.