BDAG has four definitions:

• the inhabited earth (as opposed to the heavens and hell)
• the "known" inhabited world (which could be the Roman Empire, but others have Greek Empire -- whatever empire you are a part of)
• the inhabitants of the earth
• (extraordinary use) the whole world including the heavens (!)

Interestingly a lot of people focus on Rome and forget all about Greece, but in many ways Greece was more important as the source of Hellenism which Rome also adopted and that spread even beyond the borders of the Roman empire (into Mesopotamia, all the way to India).

This interpretations of "world" and "earth" is something moderns struggle with because in the past there was less specific interest in political geography than we expect. See especially debates about Kosmos. Usually when these words are used, the main point is not areas of a map but there is some other meaning intended which leaves many readers unsatisfied, as they insist on knowing what area of the map is being colored by the term.

The only solution to this problem is to stop trying to think like a modern and take the text for what it does tell you rather than for whatever questions you bring to it.

Specifically with Matthew 24:14:

And this gospel of the kingdom will be proclaimed in the whole inhabited earth for a testimony to all the nations, and then the end will come. (LEB)

We should look at the parallelism and conclude that the gospel will be preached to all the parts of the earth that are inhabited by people so that it will be preached to all the nations. I would not try to paint a specific geographic area and say that it means one empire or another.

Source

οἰκουμένη, ης, ἡ (the pres. fem. part. pass. of οἰκέω; sc. γῆ. Hdt.+; loanw. in rabb.). ① the earth as inhabited area, exclusive of the heavens above and nether regions, the inhabited earth, the world (Ps 23:1 and often; Iren., Orig., Hippol., Did., Theoph. Ant.): πάσας τ. βασιλείας τ. οἰκουμένης Lk 4:5. Cp. 21:26; Ro 10:18 (Ps 18:5); Hb 1:6 (but s. FSchierse, Verheissung u. Heilsvollendung ’55: ‘heavenly realm’). ὅλη ἡ οἰκ. the whole inhabited earth (Diod S 12, 2, 1 καθʼ ὅλην τὴν οἰκουμένην; EpArist 37.—Diod S 3, 64, 6 and Jos., Bell. 7, 43 πᾶσα ἡ οἰκ.) Mt 24:14; Ac 11:28; Rv 3:10; 16:14; GJs 4:1. W. πᾶσα as qualifier AcPl Ha 9, 5. οἱ κατὰ τὴν οἰκ. ἄνθρωποι PtK 15:20. αἱ κατὰ τὴν οἰκ. ἐκκλησίαι the congregations throughout the world MPol 5:1; cp. 8:1; 19:2. ② the world as administrative unit, the Roman Empire (in the hyperbolic diction commonly used in ref. to emperors, the Rom. Emp. equalled the whole world [as e.g. Xerxes’ empire: Ael. Aristid. 54 p. 675 D., and of Cyrus: Jos., Ant. 11, 3]: OGI 666, 4; 668, 5 τῷ σωτῆρι κ. εεργέτῃ τῆς οἰκουμένης [Nero]; 669, 10; SIG 906 A, 3f τὸν πάσης οἰκουμένης δεσπότην [Julian]; cp. Artapanus: 726 Fgm. 3, 22 Jac., in Eus., PE 9, 27, 22: God as ὁ τῆς οἰκ. δεσπότης; POxy 1021, 5ff; Sb 176, 2.—Cp. 1 Esdr 2:2; Philo, Leg. ad Gai. 16; Jos., Bell. 4, 656, Ant. 19, 193; Just., A I, 27, 2; Ath. 1, 1) ⓐ as such Ac 24:5 (as Jos., Ant. 12, 48 πᾶσι τοῖς κατὰ τὴν οἰκουμένην Ἰουδαίοις, except that οἰκ. here is used in the sense of 1 above as surface area. Cp. PLond VI, 1912, 100). ⓑ its inhabitants 17:6.—GAalders, Het Romeinsche Imperium en het NT ’38. ③ all inhabitants of the earth, fig. extension of 1 (cp. γῆ) : world, humankind Ac 17:31 (cp. Ps 9:9; Artapanus: 726 Fgm. 3:22 Jac., God as ὁ τῆς οἰκ. δεσπότης); 19:27. Of Satan: ὁ πλανῶν τὴν οἰκ. ὅλην who deceives all humankind Rv 12:9. The passage ἐξῆλθεν δόγμα παρὰ Καίσαρος Αὐγούστου ἀπογράφεσθαι πᾶσαν τὴν οἰκουμένην (cp. κόσμον … πάντα LBW II, 1192, 6) Lk 2:1 belongs here also. For the evangelist considers it of great importance that the birth of the world’s savior coincided w. another event that also affected every person in the ‘world’. But it can also be said of Augustus that he ruled the οἰκ., because the word is used also in the sense of 2 above. For connection of the birth of a ruler with the fortunes of humanity s. IPriene 105, 4–11 on the significance of the birth of Caesar Augustus. ④ an extraordinary use: τὴν οἰκ. ἔκτισας 1 Cl 60:1, where οἰκ. seems to mean the whole world (so far as living beings inhabit it, therefore the realm of transcendent beings as well). S. Johnston s.v. κόσμος.—Also ἡ οἰκ. ἡ μέλλουσα Hb 2:5=ὁ μέλλων αἰών (6:5); JMeier, Biblica 66, ’85, 504–33; s. αἰών 2b.—JKaerst, Die antike Idee der Oekumene 1903; JVogt, Orbis terrarum 1929; MPaeslack, Theologia Viatorum II, ’50, 33–47.—GHusson, ΟΙΚΙΑ: Le vocabulaire de la maison privée en Égypte d’après les papyrus Grecs ’83; Pauly-W. XVII 2123–74; Kl. Pauly IV 254–56; B. 13.—DELG s.v. οἶκο C. M-M. EDNT. TW. Sv.

Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., pp. 699–700). Chicago: University of Chicago Press.

κόσμος, ου, ὁ (Hom.+) ① that which serves to beautify through decoration, adornment, adorning ... ② condition of orderliness, orderly arrangement, order ... ③ the sum total of everything here and now, the world, the (orderly) universe, ... ④ the sum total of all beings above the level of the animals, the world, ... ⑤ planet earth as a place of inhabitation, the world ... ⑥ humanity in general, the world ... ⑦ the system of human existence in its many aspects, the world ... ⑧ collective aspect of an entity, totality, sum total ... -- Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). In A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 561-563). University of Chicago Press.

The word "world" (κοσμος) can have a number of meanings other than "every human being without exception," which is how it is often taken. A.W. Pink maintained that there are 7 uses for the word "world" and John Own offered (I believe) 16. In John's gospel there are a few meanings present, ranging from "every human being without exception" to "all nations of the earth" to "non-elect humanity." In John 3:16, I believe John is referring to this first use, reflecting a real love of God for all that he has made, and to the universal call to repentance and faith in Christ. In John 17:9, however, he is speaking about the world which does not (and will not–John 17:20) believe in him for salvation, or "non-elect humanity."

The importance of this is that Jesus intercession and prayer are tied to his sacrifice. As the High Priest of our confession (Hebrews 3:1), his prayers for his own will be as effective as his sacrifice for them. Jesus prays for his own, and not the world, and his prayer is heard.

To directly answer the question, I'd just say that we reconcile these two seemingly contradictory statements in John by examining their contexts. Upon doing so, I think the answer that emerges is that in John 3, Jesus is speaking about God's love for all mankind which moved him to send his Son as a real offer of love and salvation to all who would receive him. In John 17, Jesus is not praying for the world, that is, those whom he knows will not receive him because they are not his; they were not given to him by the Father and in that sense do not belong to him (cf. John 10).