(Surely there are one or two of you out there who don't text?)
A text message is a communication between two cellular telephones (or something spoofing as a cellular phone), or between a commercial sender and a cellular phone. This means you don't see a sender identified per se on a text message, or some equivalent of Caller ID. What you will see on a text message is the originating phone number (ie, 8885551212) in the case of another phone, or in the case of a commercial message, a five- or six-digit code that functions as a phone number equivalent, though they are undialable.
Most phones check the originating number against the internal address book and supply whatever name is in the address book. So, if I send you a message, and you don't have me in your phone directory, all you'll see is the originating number. Which you may or may not recognize, depending on how frequently we communicate.
If you have me in your phone directory under a nickname, say "clownfeet", that's what will show as the sender. It would only have my name if your directory had my name. Likewise, I could send the same message to you, then to someone else, and you might each see a different sender based on how you do (or don't) have me listed in your phone's directory.
The narrow technical way of looking at this is the text messages move via a protocol called "SMPP", which is in a sense roughly analogous to email's SMTP protocol, where the sender ID is always the originating cell phone number (or commercial short code). In email, a name field is usually associated with the actual address, so you'll see things like this:
"Jay Lake" <firstname.lastname@example.org>
But there is no name field in SMPP, just the originating number — it's a very stripped down protocol.
In other words, the header on the message will not state who it is from, it will only state the originating phone number. However, the user's phone will do a best-effort to provide a sender name if one is available in the internal address book.
Possibly sources of confusion from this analysis include:
1) Someone else picks up the sending phone and uses it. the_child occasionally texts from either my phone or her mother's, but the recipient has no way to know that it is her sending the message instead of the phone's owner, unless she identifies herself within the message body. Or from context, of course. In story terms as in real life, this leaves many opportunities for social engineering if someone is careless with their cell phone, or has misplaced their trust in an associate.
2) All the major carriers operate email gateways, so you can send me a text at email@example.com, for example. [Not my real number, obviously.] This has a tendency to munge the sender ID in the header, as the message is not originating from another cell phone and the gateways are not especially sophisticated. This is true even if the message was emailed from another cell phone's email client or Web browser, the point being it arrived on the carrier network via SMTP rather than SMPP, and was translated by the gateway. So it's possible to have legitimate messages from a known sender with munged headers, resulting in no intelligible sender information. In story terms, one might use this method to spoof the source of the message, for example, by relying on the obscuring effect of the gateway.
3) It is also possible under some circumstances for a bad actor to spoof either the header or the entire message. This takes a lot of know-how, some special equipment and a willingness to commit felonies under Federal law. For virtually all ordinary purposes, this can be disregarded, but be aware it can be done. Mostly only fraud management and security types care about this, given that there can be a presumption of identification associate with cell phone use, that occasionally plays into dual-factor security regimes. The story applications of this should be obvious.
Grammar, spelling and punctuation in text messages
There's a tendency to assume that punctuation and whatnot is optional in text. Experience certainly might suggest that. But the protocol is content-agnostic, meaning you can send any punctuation you want. There's a habit, especially among teens and young adults, of leaving punctuation out, but that's a cultural/social marker, not inherent in the mechanics of texting. Note, however, on an older phone or dumbphone with no keyboard capability, the tediousness of multi-tap entry can drive even language pedants to such shortcuts.
Most of what we do see as text messaging usage derives from leetspeek. Wiki explains it much better than I can, but it's very much worth paying attention to this if you're using text messaging as in-story dialog. This becomes an issue of character speech register and dialect, just like any other form of dialog. For example, the_child's texts tend to use a lot of shortcuts, mine tend to be fully spelled out with no abbreviations, though I've been known to skimp on the punctuation.
Note also that there's a 140-character limit to text messages which tends to inflect some people's use of abbreviations, even those who by generational cohort or social self-selection might normally not do so.
The key point here is that the diction is in fact independent of the technology, and is largely derived from online chat, chatrooms, etc. As such, it is very dependent on the demographics and psychographics of the user.
If that was of interest to you and raises further questions, or you want a deeper dive into the technical and business side for some reason, as always please feel free to ask away in comments.