ZeroGravitas
The Destroyer
14 Segments: These allow all capital letters to be clearly displayed and add clarifying embellishments to a couple of the numbers.
The main difficulty with wiring these is routing to the central (diagonal and vertical) segments, requiring a gap of at least one tile between the enclosing segments. And that's if the wire colours are chosen very carefully! You could probably slim down the width of a display from 16 to the minimum possible 14 tiles width (for a loom of 14 control wires), but it would come at the expense of readability, with bigger (looking) gaps between smaller segments.
to make it a plain flat piece when there are other rail sections around it in every direction! So it's best to keep a 1 tile row free between each data reading rail.
16 Segments: split the top and bottom outer horizontal segments into 2 halves. This is my best shot at 16 segs. I think it's chunky components 'sexier', but I couldn't find a good wiring to keep all the gaps small, and the one wire too much pushes the width of this unit out even further (to 20 tiles wide).There is little advantage to this format unless you are intent on displaying lower case letters too. And these are inherently less readable anyway.
Reference of all letters and numbers for a 16 seg display.ZeroGravitas's Minecart World Demonstration: For compactness I settled upon 14 segment displays for my latest video (to get more than 4 characters on screen at the same time). This demonstration shows that minecarts work ok to uptade the displays with a series of fixed messages. The displays are blanked (ready for the letters on the next row) by running the cart back over the same line. It can run in either direction, as long as the same plates are all hit again
Gemspark blocks natively turn and an off with just a wiring running through them. I actuated them too to increase the visual effect, boosting the contrast between on and off. Make sure to get the phases of these 2 effects matched together, or they will mostly cancle out and look weak. Also, by selecting a background wall of the right colour, the disabled segments are better hidden. They may then be entirely invisible in map view (making it look neat there at least).
Since I was writing a fairly large message, I thought it easiest to construct a full look-up table of all letters and characters. This allowed me to simply copy-paste the related track section for each letter (in TEdit), to compose the message. If you want to use my wiring layout, you can take a copy of this table from my: "Minecart World" V0.1 download. The wiring order of the segments does not conform to any known standard, I was just happy to be able to connect to them at all!
'Cartronics' vs Hoiktronics: A (waist height bottom) hoik run would only be a little quicker, at 60tiles/s compared to 48tiles/s, and would take up more vertical space with far more complex, less accessible construction. However, minecarts are obviously limited to 1, while a hoiktronic approach could run several 'actors' to update displays simultaneously. Hoiktronic reset lines could probably be more compact too, if programming more flexible display outputs. However, I feel that the number of control lines (and physical space!) needed for each letter will probably prove inhibitive to text based output from anything resembling a programmable computer, in Terraria. These may be limited to novelty displays. They can be constructed (at great effort) to hold relatively well hidden messages, since they can only easily be deciphered by running and entering the world. (Currently you can't even see the difference between plates and plain track in TEdit).
Previous Implementations: all credit to 'SlowNick', who made a (single) 16 segment display in Terraria back in 2012, with only one wire colour! it uses lava behind active blocks (no pretty gemspark either) and flashes up letters sequentially (which is something you could do if copy/pasting a series of displays is not an option for you). It uses boulders to change the states, but parallel bird lines trigger the segments shown briefly in his "Keyboard with LCD" vid.
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It's buried in the comments here and this thread isn't titled to include matrix displays. So by all means copy paste it out and just leave a place-holder link in the comment space above this, or whatever. More people are likely to see it then, etc. Secondly, sorry it's taken me quite a while to reply here.