I've been looking at it further and it seems that the tables are named incorrectly?
My DB index is 'R43my' starts with capital letter. However, I've discovered that in the DB the tables are named as following:
r43my_relevanssi
R43my_relevanssi_cache
R43my_relevanssi_excerpt_cache
R43my_relevanssi_log
R43my_relevanssi_stopwords
Please note that the table r43my_relevanssi is named with all the small letters. I wonder if it breaks before of that.
Hopefully the plugin author can look into it.
Many thanks, Dasha