All sources have their agendas. RCC historians have revised history according to their own agenda.
I am not sure what bias or what color of eye-glass the book you refer is looking through. No doubt it is not completely objective. It has a purpose for being written.
Here is what Smith writes, from the Smith Bible Dictionary:
He (Timothy) must have joined the apostle, however, apparently soon after his arrival at Rome, and was with him when the Epistles to the Philippians, to the Colossians and to Philemon were written. Phi_1:1; Phi_2:19; Col_1:1; Phm_1:1. All the indications of this period, point to incessant missionary activity. From the two Epistles addressed to Timothy, we are able to put together a few notices as to his later from 1Ti_1:3, that he and his master, after the release of the latter, from his imprisonment, A.D. 63, revisited proconsular Asia; that the apostle , then continued his Journey to Macedonia, while the disciple remained, half reluctantly, even weeping at the separation, 2Ti_1:4, at Ephesus, to check, if possible, the outgrowth of heresy and licentiousness which had sprung up there.
The position in which he found himself might well make him anxious. He used to rule presbyters, most of whom were older than himself 1Ti_4:12. Leaders of rival sects were there. The name of his beloved teacher was no longer honored as it had been. We cannot wonder that the apostle, knowing these trials should be full of anxiety and fear for his disciple's steadfastness. In the Second Epistle to him, A.D. 67 or 68, this deep personal feeling utters itself yet more fully.
The last recorded words of the apostle express the earnest hope, repented yet more earnestly, that he might see him once again. 2Ti_4:9; 2Ti_4:21. We may hazard the conjecture that, he reached him in time, and that the last hours of the teacher were soothed, by the presence of the disciple, whom he loved so truly. Some writers have seen in Heb_13:23, an indication that he even shared St. Paul's imprisonment, and was released from i, t by the death of Nero.
Beyond this, all is apocryphal and uncertain. He continued, according to the old traditions, to act as bishop of Ephesus, and died a martyr's death, under Domitian or Nerva. A somewhat startling theory as to the intervening period of his life has found favor with some. If he continued, according to the received tradition, to be bishop of Ephesus, then he, and no other, must have been the "angel" of the church of Ephesus, to whom the message of Rev_2:1-7 was addressed.
There is no reason not to believe the above account in that it accords with Scripture.