Notice the different emphasis:Ἄξιον δὴ ἐνταῦθα γενομένους διερευνῆσαι καὶ τῶν ἄλλων τῶν μετὰ Μωσέα παρὰ τοῖς Ἑβραίοις προφητῶν τοὺς χρόνους. Μετὰ τὴν Μωυσέως τοῦ βίου τελευτὴν διαδέχεται τὴν ἡγεμονίαν τοῦ λαοῦ Ἰησοῦς πολεμῶν μὲν ἔτη ξεʹ, ἐν δὲ τῇ γῇ τῇ ἀγαθῇ ἄλλα πέντε καὶ εἴκοσι ἀναπαυσάμενος. ὡς δὲ τὸ βιβλίον τοῦ Ἰησοῦ περιέχει, διεδέξατο τὸν Μωυσέα ὁ προειρημένος ἀνὴρ ἔτη κζʹ. ἔπειτα ἁμαρτόντες οἱ Ἑβραῖοι παραδίδονται Χουσαχὰρ βασιλεῖ Μεσοποταμίας ἔτεσιν ὀκτώ, ὡς ἡ τῶν Κριτῶν ἱστορεῖ βίβλος
It is worth our while, having reached this point, to examine the dates of the other prophets among the Hebrews who succeeded Moses. After the close of Moses's life, Joshua succeeded to the leadership of the people, and he, after warring for sixty-five years, rested in the good land other five-and-twenty. As the book of Joshua relates, the above mentioned man was the successor of Moses twenty-seven years. (Strom 1.21.109.1 - 5)
Notice the difference between the original material cited (= 450 years) and the "proposed" (προκειμένη) chronology juxtaposed against it. All that we can be certain of from the original chronology is that it was counted "from" (ἀπὸ) Joshua, not necessarily his rule.γίνονται οὖν ἀπὸ Ἰησοῦ τοῦ Ναυῆ ἕως παρέλαβε τὴν βασιλείαν ∆αβίδ, ὡς μέν τινες, ἔτη τετρακόσια πεντήκοντα,
From Joshua the son of Nun, then, till David received the kingdom, there intervene, according to some, four hundred and fifty years.
... but as the proposed chronology shows, five hundred and twenty-three years and seven months are comprehended till the death of David.
... ὡς δὲ ἡ προκειμένη δείκνυσι χρονογραφία, συνάγονται ἔτη πεντακόσια εἴκοσι τρία μῆνες ἑπτὰ εἰς τὴν τοῦ ∆αβὶδ τελευτήν. (1.21.113.1)
We this in what immediately follows:
καὶ μετὰ ταῦτα ἐβασίλευσε Σολομὼν υἱὸς ∆αβὶδ ἔτη τεσσαράκοντα. διαμένει ἐπὶ τούτου Νάθαν προφητεύων, ὃς καὶ παρεκάλει αὐτὸν περὶ τῆς τοῦ ναοῦ οἰκοδομῆς· ὁμοίως καὶ Ἀχίας ἐκ Σηλὼμ προφητεύει, ἦσαν δὲ καὶ οἱ βασιλεῖς ἄμφω, ὅ τε ∆αβὶδ ὅ τε Σολομών, προφῆται. Σαδὼκ δὲ ὁ ἀρχιερεὺς πρῶτος ἐν τῷ ναῷ, ὃν ᾠκοδόμησε Σολομών, ἱεράτευσεν, ὄγδοος ὢν ἀπὸ Ἀαρὼν τοῦ πρώτου ἀρχιερέως. γίνονται οὖν ἀπὸ Μωυσέως ἐπὶ τὴν Σολομῶνος ἡλικίαν, ὡς μέν τινές φασιν, ἔτη πεντακόσια ἐνενήκοντα πέντε, ὡς δὲ ἕτεροι, πεντακόσια ἑβδομήκοντα ἕξ.
And after this Solomon the son of David reigned forty years. Under him Nathan continued to prophesy, who also exhorted him respecting the building of the temple. Achias of Shilo also prophesied. And both the kings, David and Solomon, were prophets. And Sadoc the high priest was the first who ministered in the temple which Solomon built, being the eighth from Aaron, the first high priest. From Moses, then, to the age of Solomon, as some say, are five hundred and ninety-five years, and as others, five hundred and seventy-six. (Stromata 1.21.113.2 - 4)
εἰ δέ τις τοῖς ἀπὸ Ἰησοῦ μέχρι ∆αβὶδ τετρακοσίοις πεντήκοντα ἔτεσι συγκαταριθμήσαι τὰ τῆς Μωυσέως στρατηγίας τεσσαράκοντα καὶ τὰ ἄλλα τὰ ὀγδοήκοντα ἔτη, ἃ γεγόνει ὁ Μωυσῆς πρὸ τοῦ τὴν ἔξοδον τοῖς Ἑβραίοις ἀπὸ Αἰγύπτου γεγονέναι, προσθείη τε τούτοις τὰ τῆς βασιλείας τῆς ∆αβὶδ τεσσαράκοντα ἔτη, συνάξει ἔτη τὰ πάντα ἑξακόσια δέκα. ἀκριβέστερον δὲ ἡ καθ' ἡμᾶς χρονογραφία πρόεισιν, εἰ τοῖς πεντακοσίοις εἴκοσι καὶ τρισὶ καὶ μησὶν ἑπτὰ μέχρι τῆς ∆αβὶδ τελευτῆς προσθείη τις τά τε τοῦ Μωυσέως ἑκατὸν εἴκοσι ἔτη τά τε τοῦ Σολομῶνος τεσσαράκοντα· συνάξει γὰρ τὰ πάντα ἐπὶ τὴν Σολομῶνος τελευτὴν ἔτη ἑξακόσια ὀγδοήκοντα τρία μῆνας ἑπτά.
And if you count, along with the four hundred and fifty years from Joshua to David, the forty years of the rule of Moses, and the other eighty years of Moses's life previous to the exodus of the Hebrews from Egypt, you will make up the sum in all of six hundred and ten years. But our chronology will run more correctly, if to the five hundred and twenty-three years and seven months till the death of David, you add the hundred and twenty years of Moses and the forty years of Solomon. For you will make up in all, down to the death of Solomon, six hundred and eighty-three years and seven months. (1.21.114.1 - 2)