In terms of identifying and labelling the populations in question, we suggest:
1. Climate refugees and climate change refugees do not fall within UNHCR's definition of "refugee". Definitions can change while countries can make their own decisions, as New Zealand implied in 2017 https://www.cgdev.org/blog/new-zealands-climate-refugee-visas-lessons-rest-world - compare to the Pacific Access Category https://www.immigration.govt.nz/new-zealand-visas/apply-for-a-visa/about-visa/pacific-access-category-resident-visa
2. Climate change migrants are hard to identify and count https://doi.org/10.3390/socsci8050131
3. Climate migrants definitely exist, such as UK and Norway residents purchasing retirement homes along the Mediterranean and Canadians spending the winter in Florida. We usually call them "ex-pats" or "snowbirds", because apparently "migrant" might be pejorative. Not that we would ever label people differently according to their origin.
4. Determining forced migration in the context of climate change is not straightforward or necessarily consistent:
(a) https://doi.org/10.24043/isj.52
(b) https://doi.org/10.24043/isj.120
(c) https://doi.org/10.1002/2014EF000278
(a) https://doi.org/10.1111/geoj.12177
(b) and Lakshadweep https://doi.org/10.1080/04353684.2017.1353888
(c) https://doi.org/10.1007/s10584-019-02376-y
(d) and St. Kitts https://doi.org/10.3390/atmos9120459