Where does each MySQL data file type belong?

I’m in the process of transferring a database from one server to another.
I’m moving the data files directly, instead of using a SQL dump. To do this, I:

• ensured that both servers use the same MySQL version
• set innodb_fast_shutdown = 0
• stopped the MySQL server
• rsynced the data to the new server

I’m almost ready to start the new server, but I have a doubt. The old server uses the same data dir /var/lib/mysql for data, binlogs, etc., whereas the new server uses 3 different dirs for data, binlogs and tmp:

datadir=/mysql/data
tmpdir=/mysql/tmp
log-bin=/mysql/binlog/mysql-bin

The question is: where should each of the files go?

My reasoning is:

• /mysql/tmp: there should not be any temporary files left after a slow (or even a normal?) shutdown, so nothing should go there right now
• /mysql/binlog: I should only move the mysql-bin.* files
• /mysql/data: everything else should go there

Is that correct?

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Here is the file list, for information (nxxxxxxx is my main database):

# ls -al /var/lib/mysql
total 38518860
drwxr-x--x.  9 mysql mysql       4096 20 nov.  13:50 .
drwxr-xr-x. 37 root  root        4096 29 déc.   2019 ..
-rw-rw----.  1 mysql mysql         56 21 déc.   2014 auto.cnf
-rw-------   1 mysql mysql       1680 27 déc.   2018 ca-key.pem
-rw-r--r--   1 mysql mysql       1112 27 déc.   2018 ca.pem
-rw-r--r--   1 mysql mysql       1112 27 déc.   2018 client-cert.pem
-rw-------   1 mysql mysql       1676 27 déc.   2018 client-key.pem
-rw-r-----   1 mysql mysql     589824 20 nov.  13:50 #ib_16384_0.dblwr
-rw-r-----   1 mysql mysql    8978432 19 nov.  18:28 #ib_16384_1.dblwr
-rw-r-----   1 mysql mysql      91522 20 nov.  13:50 ib_buffer_pool
-rw-r-----.  1 mysql mysql 9143582720 20 nov.  13:50 ibdata1
-rw-r-----   1 mysql mysql 1073741824 20 nov.  13:50 ib_logfile0
-rw-r-----   1 mysql mysql 1073741824 20 nov.  13:50 ib_logfile1
drwxr-x---   2 mysql mysql       4096 20 nov.  13:50 #innodb_temp
drwxr-sr-x.  2 mysql mysql       4096 19 nov.  18:28 mysql
-rw-r-----   1 mysql mysql 1073742002  5 nov.  21:45 mysql-bin.002418
-rw-r-----   1 mysql mysql 1073742033  6 nov.  12:00 mysql-bin.002419
                                    (...)
-rw-r-----   1 mysql mysql     341943 20 nov.  13:50 mysql-bin.002446
-rw-r-----   1 mysql mysql        179 20 nov.  13:50 mysql-bin.002447
-rw-r-----   1 mysql mysql        570 20 nov.  13:50 mysql-bin.index
-rw-r-----   1 mysql mysql   62914560 20 nov.  13:50 mysql.ibd
-rw-r--r--.  1 mysql mysql          6 19 nov.  18:28 mysql_upgrade_info
drwxr-sr-x.  2 mysql mysql      69632 17 nov.  17:15 nxxxxxxx
drwxr-x---   2 mysql mysql       4096 19 nov.  18:28 performance_schema
-rw-------   1 mysql mysql       1676 27 déc.   2018 private_key.pem
-rw-r--r--   1 mysql mysql        452 27 déc.   2018 public_key.pem
-rw-r--r--   1 mysql mysql       1112 27 déc.   2018 server-cert.pem
-rw-------   1 mysql mysql       1680 27 déc.   2018 server-key.pem
drwxr-x---.  2 mysql mysql      12288 27 déc.   2018 sys
drwxr-x---   2 mysql mysql       4096  1 mars   2019 test
-rw-r-----   1 mysql mysql  109051904 20 nov.  13:50 undo_001
-rw-r-----   1 mysql mysql  109051904 20 nov.  13:50 undo_002