How do I send 'fake' chat?

Note: This feature will only work in a LocalScript (or a ModuleScript running on the client)

Note: EVERYBODY will see this message.

What you're talking about is making a chat message via the core gui's.

As it says in the Wiki, "SetCore" , you need to do the following.

game:GetService("StarterGui"):SetCore("ChatMakeSystemMessage",{ --Getting "StarterGui" and calling the function
    Text = "Welcome to my game!"; -- Required. Has to be a string! Basically what the chat says
    Color = Color3.new(0,1,1); -- Cyan is (0,255/255,255/255) -- Optional defaults to white: Color3.new(255/255, 255/255, 243/255) If you require a full set http://snipplr.com/view/40782/color-to-rgb-value-table-in-lua/
    Font = Enum.Font.SourceSans; -- Optional, defaults to Enum.Font.SourceSansBold
    FontSize = Enum.FontSize.Size24; -- Optional, defaults to Enum.FontSize.Size18
})

For example;

game:GetService("StarterGui"):SetCore("ChatMakeSystemMessage",{ --Getting "StarterGui" and calling the function
    Text = "Your data has been saved";
    Color = Color3.new(0/255, 128/255, 0/255); 
    Font = Enum.Font.SourceSans;
    FontSize = Enum.FontSize.Size24;
})

This would print out like; As you said, https://gyazo.com/51c478f43e58a1ea842dd4db38967bf0

Currently, SetCore has the following options,

PointsNotificationsActive: boolean: whether or not a notification should be shown when a player gains or loses playerpoints

BadgesNotificationsActive: boolean: whether or not a notification should be shown when a player earns a badge

ChatWindowSize: UDim2: Sets the size of the chat window. Note: This size does not include the chat bar.

ChatWindowPosition: UDim2: Sets the position of the chat window.

ChatBarDisabled: boolean: If set to true, the chatbar will be hidden, effectively muting the player.

ChatMakeSystemMessage: table: Allows you to add messages to the chat (like the help message when you join a game)

SendNotification: table: Makes a notification pop up

I hope I helped!

You can use SetCore

SetCore(
    string CoreName,
    var Parameters
) -> void

You want to use ChatMakeSystemMessage as the CoreName. The valid parameters for ChatMakeSystemMessage are poorly documented, but the example should provide what you need.

Example:

game:GetService("StarterGui"):SetCore("ChatMakeSystemMessage",{
    Text = "Welcome to my game!"; -- Required. Has to be a string!
    Color = Color3.new(0,1,1); -- Cyan is (0,255/255,255/255) -- Optional defaults to white: Color3.new(255/255, 255/255, 243/255)
    Font = Enum.Font.SourceSans; -- Optional, defaults to Enum.Font.SourceSansBold
    FontSize = Enum.FontSize.Size24; -- Optional, defaults to Enum.FontSize.Size18
})