始まる and 始める: start times and the person starting
Compare an event's start time with someone starting an activity, focusing on standalone verbs and the difference between before and after the start.
When an event starts
説明会は十時に始まります。 announces that the information session starts at ten. With 始まる, read はじまる, the event itself is the topic or subject.
説明会が始まる前に、受付を済ませてください。 asks participants to complete registration before the session starts. 始まる前に places the action before the start.
A person starts an activity
全員そろったので、司会者が説明を始めました。 says that the host began the explanation once everyone was present. With 始める, read はじめる, を marks the activity begun.
去年、日本語の勉強を始めました。 says that the speaker began studying Japanese last year. The study is a noun object; additional context tells us whether it continues now.
Something has already started
着いたときには、映画はもう始まっていました。 means that the film had already started when the speaker arrived. Do not reverse the order and place the start after arrival.
始めましょう proposes starting an activity together. Do not simply replace it with 始まりましょう; identify from context what the participants are about to start.
Which happened first in 着いたときには映画はもう始まっていました: the film starting or the arrival?
Show the answer and why
The film started first. もう始まっていました presents it as already started at the time of arrival.