I read this question the day before, and I agreed strongly and didn't have any concrete responses to your questions. However, having let it simmer for a bit, I (slightly) less strongly agree that advocacy is a better term than welfare + have some half-baked responses to your questions.

Are there other considerations regarding which term to use?

I think it's valuable to also think about if the word being used is "person-centered" or "other-centered." I have noticed that people respond much more positively to the latter than the former. For instance, it's easier to get people to sign a petition that ~aligns with their values vs. making them change their day-to-day actions. It is natural to be change averse, so a solution where the hard work can be deferred to someone else will always get more traction (I am not arguing that behavior change shouldn't be done; in fact, I would much prefer a world where everyone carefully thought about ethics, but that's a separate conversation). 

This is my only reservation against the term advocacy. Historically, advocacy has been very "person-centered," so — 

• It may give newcomers/passerbys the impression that EA/adj-funded animal work is more concerned with traditional forms of advocacy and may discourage them from being involved further. 
  • On the contrary, people who would have an averse reaction to advocacy wouldn't fall in the category of "people who would otherwise join our effort," so it doesn't matter.
• My guess is that most of the funding in this space goes into welfare reforms rather than grassroots advocacy (fact check me); if this is true, then using advocacy could be a bit misleading.

All that being said, I still think Animal Advocacy is a much better umbrella term than welfare.

Is there a third term that's better than both of them?

I am unsure if any of these are better, but here's a list:

• Animal Advocacy and Reform
• [Something] Reform
  • I like reform a lot; it's well-received by most and gives them the impression that changes being proposed are incremental. 
  • However, this could alienate abolitionists further, which would be bad because they are likely to be highly-engaged in this space compared to the average person.
• (Farm/Wild) Animal Wellbeing
  • I feel this fits well and casts a wide enough net to include the different approaches being taken to solve the problem. It is also good from a promotional perspective: who doesn't want to be pro-wellbeing?
  • I also like how Wild Animal Wellbeing abbreviates to WAW.

(The English language has a poor palette of words!)

Which one of the two should EA use, and on which occasions?

Overall, I do think Animal Advocacy > Animal Welfare, so as a rule of thumb, it is probably better to use the former in most circumstances. I am in favor of the topic tag being changed, but I am unsure if it's too much of a headache to do that on the backend.