All this time I thought I had this figured out. I had installed MySQL 5.0.17 before but only recently I noticed that in phpMyAdmin it shows MySQL - 5.0.17-nt-log but MySQL client version: 4.1.7. Whats the difference between the two ? Yet, on my server it shows Client API version 5.0.20a yet MySQL - 4.0.24_Debian-10sarge1 in phpMyAdmin.
For both of your examples, the "Client" refers to the library that has been linked with PHP (phpMyAdmin might supply its own, I do not know if it does) to make the function calls to mysql. The version of the client is separate from the version of the mysql server software.
For your first example, you have version 5 of mysql. For your second example, you have version 4 of mysql.
As long as the client version is new enough, it can make calls to any version of mysql server. I don't know any of the specifics of what version of client works with what version of the mysql server software.