Apple - When does Messages message via iMessage vs. SMS?
To send SMS text messages I use the green callout balloon button that's near the phone button.
You are using the correct app.
My colleague said I could save money by using iMessage instead of paying for each SMS text message.
Your colleague is absolutely right. At least in this context. iMessage is Apple's Internet messaging service. You can send an iMessage to anyone who has an Apple device (such as iPhone, iPod Touch, iPad, Apple Watch, Mac desktop or laptop computer etc.). The requirements is that the recipient must have registered their Apple ID with iMessage service and enabled iMessage. Either of email address (Apple ID) or mobile number (associated with your iPhone) can be registered with iMessage and can be used as sending destination.
The same Messages app is used for sending both an iMessage and SMS to contacts and recipients. An SMS is identified with a green bubble while an iMessage is identified with a blue bubble.
Additionally, you need not remain online all the time for iMessage to work, if either the sender of receiver is offline (not connected to the Internet), the sent message gets stored on device or Apple's servers respectively, and dispatched as soon as the Internet connectivity is restored (either via Mobile Data or Wi-Fi). It in a way works for how email works. Unlike some other messaging app, there's no concept of user being "online" or "offline" in iMessage.
When the recipient is either using a non iPhone phone or they don't have their number registered with iMessage service, your text message gets sent as a plain SMS, and all the same restrictions apply (character count, carrier charges). However, iMessage allows you to send text messages, media such as photos and videos and with iMessage apps stickers etc.
It's important to note that iMessage is a service available free of cost and is exclusive to Apple ecosystem of devices.
You can force an iMessage to be sent as a plain SMS if you suspect the recipient may have their Internet connection (Mobile Data or Wi-Fi) turned off, and you want the message to get across as quickly as possible (as an SMS is sent over cellular network).
Is it possible to explain in a few clear sentences how the messaging functionality works on my iPhone6 and when internet is used and when traditional SMS messages are used?
Answers below inline:
Does it check first for an active internet connection on the sender's phone?
No. Sending iMessage has nothing to do with active connection on recipients end.
Does it also check if the recipient is also connected to the internet?
No. The recipient gets the message whenever they turn on Internet on their device. The messages you sent are held on Apple's computers in the meanwhile.
Could a message go out as an iMessage and then not be delivered even though the recipient could have received an SMS text message?
If a message is sent out as iMessage, it will indeed get delivered (as long as their number is registered with the iMessage service) when the recipient turns on Internet on their device.
P.S.: I can direct you to this excellent document online officially maintained by Apple, which explains iMessage service and it's features in an easy to understand way.
Messages for iPhone, iPad, Apple Watch, and Mac
Force Messages app to send a text message as SMS instead of an iMessage
Here's a workaround that you can use to force the Messages app to send a test message as SMS instead of iMessage to a contact's phone number. It is slight clumsy, but gets the job done in the scenarios like you described:
If you have Internet access available on your iPhone via either Mobile data or Wi-Fi (indicated by the presence of appropriate icon in the iPhone's status bar), and the receivers contact number is registered with iMessage, you'll only be able to send an iMessage, irrespective of the online/offline status of the recipient. You can check if a message would be sent as an iMessage by the presence of a blue colored upward facing arrow besides your typed text message.
To force the message to be sent as SMS instead, temporarily turn off Mobile Data/Wi-Fi on your iPhone to disable Internet access. You'll still only be able to send an iMessage (Blue arrow icon) as your iPhone knows that the receiver is registered with iMessage. Send the message normally.
The message won't get sent from your iPhone due to the absence of an Internet connection. Now tap an hold on the message bubble, and you'll see an alert at the bottom of the screen with an option that says Send as Text Message. This option is not shown when you have Internet connectivity on your iPhone, or the message is already delivered to the Apple's servers.
This will cause your iPhone to re-send the message as SMS. The receiver will be able to get the message even in absence of the Internet connection on their device (as long as they have cellular connectivity on their device).
Don't forget to turn Mobile Data/Wi-Fi back on on your device to regain access to Internet.