If eight people plus the chazan had already finished Shemoneh Esrei, they could have started without your active participation.[1] The main source for this assertion is the Shulchan Aruch[2] (in the context of Kaddish), who rules that someone who is sleeping or cannot answer because he is in the midst of Shemoneh Esrei can count as one of the ten people needed for a minyan. This is derived from the Hagahot Maimoniot[3] and is based on opinions in the gemara[4] that are lenient regarding the qualifications of the tenth person for a minyan.
This principle is not unanimous, however. On the lenient side, some Acharonim[5] maintain that it suffices for there to be six people (including the chazan) actively involved in the Kaddish. Some distinguish between the case of sleep, in which case only one person can be counted, and a situation in which some people are reciting Shemoneh Esrei, in which case more can be counted towards the minyan, as they are involved in sanctifying HaShem and can stop their tefilla to at least listen. The Mishna Berura[6] is inconclusive on the matter. In the other direction, the Shulchan Aruch HaRav[7] holds that for chazarat hashatz, as opposed to Kaddish and most other matters that require a minyan, there must be nine people who actually respond to the chazan; neither one who is sleeping nor one who is davening counts.
Does someone who is starting with the chazan count as being involved in chazarat hashatz, or is he like anyone else who is in the midst of Shemoneh Esrei? It is cited in the names of Rav S.Z. Auerbach[8] and Rav Ovadia Yosef[9] that one who is reading along word for word with the chazan counts fully toward the minyan, as doing so is no worse than listening and answering amen.
There is a classical source and application of this concept about a person who is reading along. One is not supposed to start Shemoneh Esrei if he will not be able to respond to Kedusha and answer amen to a few critical berachot.[10] However, he may recite these passages along with the chazan.[11] This is because reciting the beracha along with the chazan is like answering.
Following this logic, it would seem that more than one such person should count toward the minyan.[12] Could we thus go so far as to say that the entire minyan can be davening Shemoneh Esrei along with the chazan? It might appear that we have a similar practice in heiche kedusha,[13] but that is conceptually different. In heiche kedusha, there is actually no chazarat hashatz, as the chazan himself does not repeat Shemoneh Esrei but says it only once. In our case, in contrast, we probably need a majority of a minyan to make chazarat hashatz meaningful, and the people who are reading along make it considered that a full minyan is participating.
We see in a related halacha that it is not crucial to have nine people answering amen; it suffices that they are involved. The Rosh,[14] who is the main source that nine people must be listening, says that all should answer amen, but he assumes the chazarat hashatz is valid if others are following along intently, even if they are not answering amen. In general, saying amen is an important mitzva, but it is not required for one to be credited with a beracha to which he is listening with the intent to fulfill a mitzva.[15] (In the context of heiche kedusha, the Rama[16] says that there should be at least one person answering.)
Therefore, had you read along with the chazan, you could have counted toward the minyan. In fact, doing that would have shown better involvement than those who were up to different sections of Shemoneh Esrei at the time. Whether it pays to wait for additional people who have finished Shemoneh Esrei before starting chazarat hashatz depends on a few variables, including the extent of tircha d’tzibbura.[17]