diff --git a/tests/data_files/early_data.txt b/tests/data_files/tls13_early_data.txt
similarity index 70%
rename from tests/data_files/early_data.txt
rename to tests/data_files/tls13_early_data.txt
index f0084c3ef0..0c84b07205 100644
--- a/tests/data_files/early_data.txt
+++ b/tests/data_files/tls13_early_data.txt
@@ -1,3 +1,3 @@
 EarlyData context: line 0                                                    lf
 EarlyData context: line 1                                                    lf
-EarlyData context: If it appear, that means early_data success
+EarlyData context: If it appears, that means early_data received.
diff --git a/tests/opt-testcases/tls13-misc.sh b/tests/opt-testcases/tls13-misc.sh
index 572a291fd2..d5efc9edc1 100755
--- a/tests/opt-testcases/tls13-misc.sh
+++ b/tests/opt-testcases/tls13-misc.sh
@@ -501,7 +501,7 @@ requires_all_configs_enabled MBEDTLS_SSL_EARLY_DATA MBEDTLS_SSL_SESSION_TICKETS
 requires_any_configs_enabled MBEDTLS_SSL_TLS1_3_KEY_EXCHANGE_MODE_PSK_ENABLED \
                              MBEDTLS_SSL_TLS1_3_KEY_EXCHANGE_MODE_PSK_EPHEMERAL_ENABLED
 run_test "TLS 1.3 G->m: EarlyData: feature is disabled, fail." \
-         "$P_SRV force_version=tls13 debug_level=4 max_early_data_size=-1 $(get_srv_psk_list)" \
+         "$P_SRV force_version=tls13 debug_level=4 max_early_data_size=-1" \
          "$G_NEXT_CLI localhost --priority=NORMAL:-VERS-ALL:+VERS-TLS1.3:+GROUP-ALL -d 10 -r --earlydata $EARLY_DATA_INPUT" \
          1 \
          -s "ClientHello: early_data(42) extension exists."                     \
diff --git a/tests/ssl-opt.sh b/tests/ssl-opt.sh
index 2ff097ff1e..696ec164aa 100755
--- a/tests/ssl-opt.sh
+++ b/tests/ssl-opt.sh
@@ -72,7 +72,7 @@ guess_config_name() {
 : ${MBEDTLS_TEST_OUTCOME_FILE=}
 : ${MBEDTLS_TEST_CONFIGURATION:="$(guess_config_name)"}
 : ${MBEDTLS_TEST_PLATFORM:="$(uname -s | tr -c \\n0-9A-Za-z _)-$(uname -m | tr -c \\n0-9A-Za-z _)"}
-: ${EARLY_DATA_INPUT:=data_files/early_data.txt}
+: ${EARLY_DATA_INPUT:=data_files/tls13_early_data.txt}
 
 O_SRV="$OPENSSL s_server -www -cert data_files/server5.crt -key data_files/server5.key"
 O_CLI="echo 'GET / HTTP/1.0' | $OPENSSL s_client"