Here's my advice

HAVE DECIDED (have + past participle) is an example of the present perfect tense.

Present perfect tense is used for an action that BEGAN in the PAST and is STILL GOING ON in the present OR an action that began in the past and IS FINISHED BY THE TIME THE SPEAKING IS DONE.

So,

When you say:

"They decided to play" you are indicating that the decisions was made and completed 100%

and

When you say:

"They have decided to play" you are indicating that they just decided or finished decided by the time you got done saying it.

It may make more sense if you switched the pronoun to I.

So if Lebron said

"I have decided to play for Miami" this indicates that his decision is made by the time he is done saying is decision. You would use this tense to indicate that the decision is hot off the press, it has just been announced.

If Lebron said,

"I decided to play for Miami" this indicates that Lebron is referencing a decision that was made, announced and aware of (IE COMPLETED) BEFORE he began speaking.

Hope this helps!