Mysql binlog filename keeps changing, preventing replication

First off, some versions: Centos 6, mysql 5.1 are what I'm stuck with using for now.

So - The problem is that when I replicate using either of these methods:

A: From the manual, unavailable via Oracle's website but thankfully mirrored by some university in Namibia: http://download.nust.na/pub6/mysql/doc/refman/5.1/en/replication.html

B: Via Percona's tools, the preferred method as it doesn't result in even a second of downtime, which is impossibly costly: https://www.percona.com/doc/percona-xtrabackup/2.3/howtos/setting_up_replication.html

That on the slave I get an error telling me that the binlog can't be found on the master. Checking the path where the binlogs are stored, the filename ends up being something different than when I start the dump or innobackupex. My mysql configuration looks like this on the master:

# cat /etc/my.cnf
[mysqld]
datadir=/var/lib/mysql
socket=/var/lib/mysql/mysql.sock
user=mysql
thread_cache_size = 60

symbolic-links=0
innodb_buffer_pool_size=6000M
#query_cache_size = 2500M
table_cache = 800M
open_files_limit = 20000
key_buffer_size=500M
log-slow-queries = /var/log/mysql_slow_query.log
long_query_time = 6
# general_log = 1
max_connections = 500
innodb_thread_concurrency = 18
join_buffer_size = 10M
#log = /var/log/mysql_queries.log
log_bin = /var/lib/mysql_dump/bin/bin
skip-name-resolve

# Replication
server-id = 1
innodb_flush_log_at_trx_commit= 1
sync_binlog= 1
max_allowed_packet=1024M

[mysqld_safe]
log-error=/var/log/mysqld.log
pid-file=/var/run/mysqld/mysqld.pid

And on the slave:

[mysqld]
datadir=/var/lib/mysql
socket=/var/lib/mysql/mysql.sock
user=mysql
# Disabling symbolic-links is recommended to prevent assorted security risks
symbolic-links=0

server-id=2

[mysqld_safe]
log-error=/var/log/mysqld.log
pid-file=/var/run/mysqld/mysqld.pid

Results of these queries as well:

mysql> select @@expire_logs_days;
+--------------------+
| @@expire_logs_days |
+--------------------+
|                  0 |
+--------------------+
1 row in set (0.00 sec)

mysql> select @@log_bin;
+-----------+
| @@log_bin |
+-----------+
|         1 |
+-----------+
1 row in set (0.00 sec)

And this is what comes up for the master status:

mysql> SHOW MASTER STATUS;
+------------+----------+--------------+------------------+
| File       | Position | Binlog_Do_DB | Binlog_Ignore_DB |
+------------+----------+--------------+------------------+
| bin.023469 | 44272212 |              |                  |
+------------+----------+--------------+------------------+
1 row in set (0.00 sec)

And on the slave:

mysql> SHOW SLAVE STATUS\G
*************************** 1. row ***************************
           Slave_IO_State: 
              Master_Host: [redacted for the public :) ]
              Master_User: repl0
              Master_Port: 3306
            Connect_Retry: 60
          Master_Log_File: bin.023468
      Read_Master_Log_Pos: 15943086
           Relay_Log_File: mysqld-relay-bin.000002
            Relay_Log_Pos: 4
    Relay_Master_Log_File: bin.023468
         Slave_IO_Running: No
        Slave_SQL_Running: Yes
          Replicate_Do_DB: 
      Replicate_Ignore_DB: 
       Replicate_Do_Table: 
   Replicate_Ignore_Table: 
  Replicate_Wild_Do_Table: 
  Replicate_Wild_Ignore_Table: 
               Last_Errno: 0
               Last_Error: 
             Skip_Counter: 0
      Exec_Master_Log_Pos: 106
          Relay_Log_Space: 106
          Until_Condition: None
           Until_Log_File: 
            Until_Log_Pos: 0
       Master_SSL_Allowed: No
       Master_SSL_CA_File: 
       Master_SSL_CA_Path: 
          Master_SSL_Cert: 
        Master_SSL_Cipher: 
        Master_SSL_Key: 
    Seconds_Behind_Master: NULL
Master_SSL_Verify_Server_Cert: No
            Last_IO_Errno: 1236
            Last_IO_Error: Got fatal error 1236 from master when reading data from binary log: 'Could not find first log file name in binary log index file'
           Last_SQL_Errno: 0
           Last_SQL_Error: 

And then this:

$ cat /var/lib/mysql/xtrabackup_binlog_info
bin.023468      15943086

It was largely the same issue when I did it Oracle's way - Simply had a different filename for the binlog than when I had done the dump.

So - how to make sure that this doesn't go wrong on my third attempt?