For the following simple setup I always get notifications for both triggers.
I would expect that a notification for "ICMP ping" would never show up.

Two triggers, "ICMP ping" depends on "ICMP ping 2":

Severity: Average
Name: ICMP ping
Expression: {host_1:icmpping.last(0)}=0
Depends on: host_1: ICMP ping 2
Status: Enabled

Severity: High
Name: ICMP ping 2
Expression: {host_1:icmpping.last(0)}=0
Status: Enabled

The action, delayed by 2 Minutes to give Zabbix time to evaluate dependencies:

Steps (sec): 2
Details: Send message to users: user (Name) via Email Start in: 00:02:00
Duration : Default

The example above is not very meaningful in practise, here another more useful example.
Even when both conditions for Trigger 3 are true, I still get notifications from all Triggers, I would expect a mail only from Trigger 3:

Host 1 -> Trigger 1 (Condition 1)
Host 1 -> Trigger 2 (Condition 2)
Host 1 -> Trigger 3 (Condition 1) and (Condition 2)

...where Trigger 1 and Trigger 2 both depend on Trigger 3.

Regards.