Sending photos/images to non Apple phones.

Your question is really a two-parter:

• What types of messages can Wi-Fi support? iMessage? (Yes.) SMS/MMS? (No.)
• What is the relationship of cellular data to SMS/MMS? (Cellular data is normally not required for SMS/MMS, as these are sent using cellular voice methods).

Regarding your first question part, see this article for more detail:

About iMessage and SMS/MMS - Apple Support

Regarding your second question part, perhaps you're using cellular data because the person(s) you're sending to used to have iMessage (Apple products) that were never de-registered properly. This may help the intended recipient(s);

Deregister iMessage on your iPhone or online - Apple Support

Joanna Tan wrote:

... those little hand-written "Hello", "Thank You" images and GIFs that iPhone has now in iMessages.

Aha!!! Perhaps that is the problem.

Those things only work with iMessage!

What I thought you were asking about was an action like sending a photo using a text message to an Android user.

By the way, since you're trying to send "the hand-written 'Hello' 'Thank you' images ...", that explains your use of cellular data when Wi-Fi is off. The thinking goes like this:

• You are sending "Hello" "Thank You" images. These images are only valid in iMessage. Therefore, you are sending an iMessage.
• iMessage normally is sent via Wi-Fi. But if Wi-Fi is not available and your cellular carrier supports iMessage, the message will be sent using cellular data.

As I said, however, MMS would not use cellular data (or Wi-Fi).

Joanna Tan wrote:

So when I am in wifi zone which is pretty much all the time, I cannot send text SMS/MMS?

No, sorry, I did not mean to suggest that conclusion at all.

I'm simply trying to say that SMS/MMS relies solely on having a cellular connection - the same type and quality of connection you would use to have a voice conversation over that cellular connection.

I only meant to imply that Wi-Fi is irrelevant to SMS/MMS.

One clarification request:

Are you able to send a simple text messages (with no images - only text) to an Android phone while cellular data is off and Wi-Fi is on? In other words, is your issue restricted solely to sending MMS messages and not SMS (simple text) messages?

And do you have the same misbehavior when receiving messages?

Are you aware that MMS often requires an extension to a basic cellular carrier plan? Does your cellular carrier know you are trying to use MMS? Often, there is some extra charge associated with this (above the SMS charge).

How are you turning off mobile data? Are you turning on Airplane Mode and then re-enabling Wi-Fi, or are you going to Settings > Cellular > Cellular Data > Off?

At the risk of stating the obvious:

Settings > Messages > MMS Messaging must be "On" to use MMS.