I accept Jason Bassford's comment that 'She was happy to' + 'verb' contains an inherent ambiguity as to which definition of 'happy' the writer intended. I will proceed on the assumption that the writer intends 'happy' to refer to 'feelings of pleasure or enjoyment' and not 'openness or willingness to do something'.
Having removed this obstacle, there remains another significant issue with the OP's question. Sentences A2 and B2, and A3 and B3 are congruent in form, i.e. the structure and tenses used in both sets of sentences are identical. The same cannot be said for sentences A1 and B1. To make B1 congruent with A1 we would need to change B1 from:
She was happy when she was winning the contest.
She was happy when she won the contest.
This now negates the OP's statement:
It seems to me that (a) corresponds to (a1), and (b) corresponds to (b2).
Now, if we accept that (a) corresponds to (a1), then we logically have to say the (b) corresponds to (b1). However, does (a) only correspond to (a1); I don't think so.
In sentences (a) and (b), the verbs 'to see' and 'to win' are both written in the infinitive form. As a consequence they do not contribute to the tense of the sentence - the infinitive of a verb is not inflected by tense. This means that we can only determine the tense of these two sentences on the basis of the finite verbs contained in each, i.e. 'was' as in 'she was happy'. As this is in the past tense, we must assume that both sentences are in the past tense.
In my opinion, because both (a) and (b) are in the past tense, there is a poor fit between them and (a3) and (b3) which both look ahead to a future event. I accept that this could be a future in the past tense, ie the event that was anticipated was in the future at the time 'she was happy', but that 'future event' could have occurred between her being happy and the present time. Nonetheless, the future aspect does not sit well with a sentence that is fully in the past tense.
In my opinion, again, (a) corresponds to both (a1) and (a2), and (b) corresponds to both (b1) and (b2). So, she was happy when she saw her mother, and continued to feel happy when she recalled having done so. Similarly, she was happy when she won the contest, and continued to feel happy when she recalled having done so.