Your examples 1b) and 2b) might or might not be interpreted to mean the same thing. Consider similar examples:

a) Bill bought a car so that he was mobile.

b) Bill bought a car so that he would be mobile.

While the second example makes it clear that Bill's purpose was to be mobile, the first example could be interpreted as either purpose or result. It's ambiguous.

However, when you omit that the meaning becomes clear. So now acts as a conjunction. Bill's promotion is the result of his hard work.

Bill worked hard so he was promoted.

To make it clear that Bill's promotion was the result of his hard work, you need to reconstruct the sentence completely.

Bill was promoted because he worked hard.

When you use so in this context, you mean as a result

When you use so that, you mean in order that - for this purpose

(https://forum.wordreference.com/threads/so-that-construction-interpretation.2826032/)

Your examples 1b) and 2b) might or might not be interpreted to mean the same thing. Consider similar examples:

a) Bill bought a car so that he was mobile.

b) Bill bought a car so that he would be mobile.

While the second example makes it clear that Bill's purpose was to be mobile, the first example could be interpreted as either purpose or result. It's ambiguous.

However, when you omit that the meaning becomes clear. So now acts as a conjunction. Bill's promotion is the result of his hard work.

Bill worked hard so he was promoted.

To make it clear that Bill's promotion was the result of his hard work, you need to reconstruct the sentence completely.

Bill was promoted because he worked hard.

When you use so in this context, you mean as a result

When you use so that, it might be interpreted as in order that (for this purpose) or as a result. It's not clear.

(https://forum.wordreference.com/threads/so-that-construction-interpretation.2826032/)