Confused about connecting smartphone to projector

Hello,

I have a projector and would like to project the screen from my many Android phones.

Unfortunately none of my phones have HDMI output.
At first I was expecting this to be as simple as connecting using a slimport/MHL micro-usb to HDMI adaptor,but I see that such adaptors are not compatible with all devices.

Then I looked at the chromecast and miracast and I see that while miracast can straight cast the home screen from the phone v/s casting from compatible apps (as with chromecast),...

Confused about connecting smartphone to projector

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