Using CONVERT TO CHARACTER SET for changing only collation?
There are a few options for managing character set conversions. As you have found, there is the alter table option, and you can use the character set clause or the collate clause or both in your statements. 

Other options include changing the character set and collation for the whole database (which I can see you don't want to do just yet). 

Or there's a tool in the free and open source Percona Toolkit that is very popular with developers, pt-online-schema-change, that helps you manage such migrations for tables with a primary key.  

David Berube, a consultant independent from Percona, wrote a detailed blog post on our community blog about some of the problems that you can encounter when changing character sets and collations. You can read that at https://www.percona.com/community-blog/2018/06/12/character-sets-migrating-utf8mb4-pt_online_schema_change/

The potential pitfalls included database version compatibility, application "expectations", key length management (the length could change), and the fact that you could see false positives i.e. a column that says it's latin1 may not be!

This time on the Percona website, there are a number of blog posts and also a free webinar Troubleshooting Issues with MySQL Character Sets https://www.percona.com/resources/webinars/troubleshooting-issues-mysql-character-sets 

Hope some of these will help you work out the best way to approach your scenarios. 

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Disclosure: I work for Percona.