How do I round a calculation involving both addition/subtraction and multiplication/division to the correct number of significant figures?
Approach 2 is better than approach 1.

This approach corresponds with methodology employed in both web links you provided.

The basic 'rule' is to do your calculation keeping all digits as you go right until to the very end (without rounding parts as you suggest in approach 1). This ensures your calculation retains maximum accuracy. You leave the 'rounding' til the very end, that is when you report your answer, only 'so many' digits are 'significant'.

To use your example, when calculating the subtraction part: 2.378-1.2 = 1.178, in approach 1, by 'rounding' this down to 2 significant digits (1.2), you are reducing the accuracy of your calculation before you finish.

The correct (most accurate) way to calculate the result is to keep all significant digits in your calculation until the very end, that is, 1.178 / 1.03 = 1.14368932 and then report only the 'significant' digits. In this case, since the number 1.2 only has 2 significant digits, your final answer can only have 2 significant digits. That is, the final answer is only as precise as the least precise measurement (in your case, 1.2), so in reporting the final answer for your calculation, you can only rely on 2 significant digits, so you report 1.1 as your calculated answer.

If you re-examine the answer provided in your link (http://www.occc.edu/kmbailey/chem1115tutorials/Sig_Fig_and_Math_Answers.htm) in which you state problem 3 follows approach 1, you will find that the solution provided in that link actually follows approach 2, similar to what I described in the above paragraph. That is, all decimal places are retained in the calculation until the very, when the final answer is reported to the number of significant figures corresponding to the least significant measurement.