in/ex-tension obviously share the same root. Did the difference in prefixes engender and explain the differences in their meanings? Can the other senses of extendo and intendo be explained based on the meanings of their prefixes?
How? To wit, how does ex- befit meanings 2-3 for extendo? Why couldn't in- have been used? What makes 'INcrease' (in 3) more ex- than in-?
How does in- befit meanings 2-4 for intendo? I'm baffled by extendo "extend, increase", and intendo "turn one's intention to, aim, intend". Why's "aim" expressed by intendo rather than extendo?
Both intendo and extendo can mean "stretch out". The meanings of the prefixes ex- and in- often differ. So how can we explain that both verbs are used in this sense? Is there a way to make sense of it, or is it best regarded as an arbitrary development?
Before answering your question(s), please let me make a couple of general remarks, which I hope can also be useful for other questions you've raised on Latin prefixes/preverbs (I've just seen you raised MANY of them in this site!):
(I) It is important to understand Latin preverbs by considering them as forming a SYSTEM (for a nice overview of the system of Latin preverbs, I recommend you to take a look at the following work by Prof. Benjamín García-Hernández (1989)). In a nutshell, you will see that Latin preverbs can be classified into different levels of analysis, providing different spatial, aspectual, or intensive meanings. So any particular or specific case you turn out to be interested in must be analyzed in relation to this system. Of course, there are many IDIOSYNCRASIES involved but it is advisable to try to analyze them in relation to the (previous) system (i.e., it is not the case that anything goes. Despite appearances, the Latin system of preverbs is not chaotic!). NB: in fact, something (very) similar happens in other so-called satellite-framed languages: e.g., cf. English phrasal verbs and Russian prefixes. For example, notice that English particles (the so-called “satellites” in Talmy's typology) also have spatial meanings (“He went out”), aspectual meanings (“It turns out to be the case…”) and intensive meanings (“He cried his eyes out”: i.e., ‘he cried a lot’). And, of course, many idiosyncrasies are also involved in English phrasal verbs.
(II) Don’t try to understand particular cases or examples of Latin preverbs ONLY by considering how they are translated into another languages (e.g., English). Translations are just translations, which do not always capture all the nuances nor even basic properties of the system (as in the present case).
With these general remarks in mind, let me address your specific question(s). You point out the curious fact that the first "meaning" (English translation!) of both intendere and extendere is “to stretch out”. So, apparently, by looking at the English translations, it could seem that, for example, bracchia intendere and bracchia extendere are fully synonymous since both can be translated as ‘to stretch out their arms’ (e.g., cf. intentaque bracchia remis (Verg. Aen. 5, 136) and extento bracchio (Cic. De orat. 242)). However, as noted above, one should not conflate meanings with translations: i.e., ‘to stretch out’ is just a "good" non-nuanced translation of both intendere and extendere but it does not capture the important difference between the illative meaning of IN- and the elative meaning of EX-. In the former we focus on the initial direction of tendere (e.g., cf. the more appropriate French translation of intendere given by Gaffiot as ‘tendre dans une direction’). In contrast, in the latter (extendere) it is the final part of the action of TEND- what is foregrounded. Notice that this contrast gives you an explanation to another question of yours above (cf. “(...) intendo ‘turn one's intention to, aim, intend'. Why's "aim" expressed by intendo rather than extendo?”). Notice that these meanings ("aim", “direct”, “turn one’s intention to”, etc.) are precisely more coherent with the INITIAL part of the action expressed by IN- rather than with the final/egressive part expressed by EX-. Similarly, notice that in Latin you can say lineam extendere 'to draw/create a line' (Plin. 9, 182) but not lineam intendere with the same meaning. The former involves focus on the creation, i.e., on the final/resultative part of the action (similarly, cf. Engl.'to make a cheque OUT'). Cf. also the initial/aspectual meaning of prefix IN- in the example primis se intendentibus tenebris (Liv. 1, 57, 8), nicely translated into French by Gaffiot as 'les ténèbres commençant à s'étendre' (i.e., lit. "with the darkness (pl.) beginning to expand').
To conclude, please notice that the previous analysis of some meaningful cases of your question can only be carried out on the basis of having the general remarks above in mind. And remember Alec Marantz’s words (a distinguished morphologist): “it’s very difficult to argue anything from idiosyncrasies –one argues from systematic differences”.
When looking at prefixed verbs like this, it's sometimes helpful to look at the prefixes as separate prepositions.
Intendō is prefixed with in, which here indicates a destination or target. Cicero talks about weapons being extended toward Rome as a threat, for example: tela intenta in patriam (D.P.C. 9, line 23 in the Loeb edition). This is how it gained the meaning "focus on", and from that the wide variety of Mediaeval meanings.
Extendō is prefixed with ex, which usually means "out of". But that doesn't make much sense here. So what gives?
Well, remember, there's always another option for prefixed verbs in Latin: whatever the prefix is, it might just be an intensifier, rather than changing the verb's semantics.
This is how both extendō and intendō can mean "stretch out": they're just intensified forms of tendō "stretch", with no major semantic differences. Similarly, it's not hard to see how extendō could be generalized from "stretch out" to "make larger".