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Which message type are you referring to

Understanding the different messaging services

• SMS (Short Message Service): Traditional text messages sent through your mobile carrier's cellular network. SMS does not require an internet connection, but you must have cellular network coverage and an active service that supports SMS.
• MMS (Multimedia Messaging Service): Used to send pictures, videos, and other multimedia content through your mobile carrier. MMS generally requires carrier support and may require cellular data, depending on your carrier and settings.
• iMessage: Apple's messaging service, which works over Wi-Fi or cellular data. It allows you to exchange messages, photos, videos, and other content with other Apple users. An internet connection is required.
• RCS (Rich Communication Services): An enhanced messaging standard that supports features such as typing indicators, read receipts, and high-quality media sharing. It uses an internet connection and requires compatible devices, software, and carrier support.
• WhatsApp, Telegram, and Discord: These are internet-based messaging platforms. They require an internet connection to send and receive messages. Their messages are handled through their respective services rather than through the traditional SMS system.

Important: Sending and receiving SMS messages does not require internet access, but iMessage, RCS, and internet-based messaging applications do. MMS requirements depend on your carrier and network configuration.

To improve my understanding of the phenomenon, it would be highly beneficial if you could share a full screenshot (Do not crop it). Visual representations carry a greater impact than written descriptions.

Please remember that this is a public forum, so when attaching a screenshot, please avoid including any personal credentials such as IP addresses, card details, email IDs, Apple IDs, IMEI numbers, serial numbers, phone numbers, order IDs, or any identifiable location information if you are sharing a map.

With the limited information that you have provided, It is hard to even guess the problems faced by you. 

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Please re-post with detail so that someone else can help.