If I'm reading this right, and I'm not sure that I am, the OP is asking about making a right turn into a bike lane on a cross street (while on a bike).
If so, auto traffic in the adjacent lanes is a non-issue, but I still take it slow, and try to time it so my intent is clear to oncoming traffic. This offers a degree of protection in case there's a surprise in the lane and I need to move left. It also reduces the chance of a driver moving into the lane for his own reasons, ie to park, or maneuver around a driver to his left.
It's not a question of law or rights but protecting myself not only from what drivers should or will do, but also from what they might do.
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