Restrictive or adjectival participal phrase?

Hi. I sometimes have difficulty seeing the difference between these sentences. Thank you for sending me an email inviting me to attend your event. Here, I think the part "inviting me to attend your event" could be an adjectival phrase that could be written like this:

Thank you for sending me an email (that is) inviting me to attend your event. (This sentence looks awkward, I might say)

Or I think it could be just a participle phrase acting restrictively (if I phrased it correctly).

I think the following is an example that shows the use of a participle phrase acting restrictlvely and not an adjectival phrase.

eg,

He was standing around pointing at the bird up in the air.