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Let's say there is a fin that is 1mm thick, extends 8mm from the surface, and is 10 mm wide. The fin is exposed to a moving fluid. Can we assume the adiabatic tip condition and use the characteristic, "Corrected", length for calculations?

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Using the corrected length does not assume an adiabatic tip. The corrected length is intended to make a non-adiabatic tip fit into equations used for an adiabatic tip. Not using the corrected length – i.e., taking $L_c=L\ $ – means assuming an adiabatic tip. –  rdhs Feb 26 '12 at 5:47
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The error due to assuming the tip is adiabatic is approximately given by the area ratio of the tip to the total surface area of the fin. That is, $error = 8 mm^2 / 160 mm^2 $.

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