Not necessarily. 1281.5(c) only applies if the case falls under 1281.5(a). If your case does not fall within that section, then general principles regarding the waiver of arbitration provisions apply. The case law on the subject is relatively extensive, and generally holds that the filing of an answer does NOT constitute a waiver of an arbitration provision. You should be able to easily find relevant cases using standard research tools.
The bank's filing of an answer by itself is not always enough to result in a waiver of an arbitration clause in a contract. But it can sometimes be, depending on the circumstances. The more that is done in the civil case, the less likely that they will be able to compel contractual arbitration.
States law differs on the issue. However under New York law (CPLR Section 75), the motion to compel must be filed within the applicable statute of limitations. If a court action has been filed, a Motion to Compel should be raised at the time the defendant files its answer.
§ 7502. Applications to the court; venue; statutes of limitation; provisional remedies
Limitation of time. If, at the time that a demand for arbitration was made or a notice of intention to arbitrate was served, the claim sought to be arbitrated would have been barred by limitation of time had it been asserted in a court of the state, a party may assert the limitation as a bar to the arbitration on an application to the court as provided in section 7503 or subdivision (b) of section 7511. The failure to assert such bar by such application shall not preclude its assertion before the arbitrators, who may, in their sole discretion, apply or not apply the bar. Except as provided in subdivision (b) of section 7511, such exercise of discretion by the arbitrators shall not be subject to review by a court on an application to confirm, vacate or modify the award.
Once service of the summons and complaint has been made it is incumbent upon the defendant to interpose a timely answer. If service of the pleadings has been personal in nature then an answer is due within twenty (20) days of receipt of the summons and complaint. In all other cases involving alternative methods of service the answer must be served within thirty (30) days of receipt of the pleadings. It is common practice in New York for plaintiffs to grant extensions of time to answer to defendants. In exchange for granting an extension of time to answer, many plaintiffs' attorneys will require that the defendant waive the affirmative defense of lack of jurisdiction and, on occasion, the affirmative defense of the Statute of Limitations.
I'm not going to give you a hard and fast Answer since I don't practice in California. (However, I will say that if you are reading that statute correctly and the Bank didn't put the arbitration agreement in its litany of affirmative defenses, it sounds like you have eased it out at the finish line.) I will give you also the answer if it happened in New York. If the defense of an arbitration agreement was not raised in the answer, it should be considered waived but it is in the discretion of a Court to allow the amendment to the Answer and a motion to dismiss and compel arbitration.
The general rule on commencing arbitration is the amount of time (absent any other agreement) in which the action could be brought; so, for instance, on a guarantee, it would be 6 years after default on the guarantee.
Notwithstanding the answer by the which failed to raise the Arbitration Agreement as a defense or a motion to stay the action and compel arbitration, motions may be made subsequently to compel arbitration. The issue before the Court in that instance is whether by participating so substantially in the action, the Bank has waived its right to arbitrate. I remember a close decision in a case where both parties failed to raise the issue of the arbitration agreement, exchanged discovery demands and documents and interrogatories and the court found that the defendant, who could have asserted the arbitration agreement earlier, lost that right by "implied waiver" as opposed to legal or intentional waiver. Given the facts as I have stated in regards to the case to which I am referring, the time period was also over 18 months. I don't know how a Court would rule on this issue today, but I should think an answer and discovery served and responded to would be enough to cause a waiver.
As you describe CCP Section 1281.2(a), California has apparently answered the question for you: you don't seem to have to arbitrate if the defense was not in the answer and you serve discovery before the Bank wakes up and attempts to Amend its Answer. Your attorney should have been able to discuss this issue with more clarity, but you may just be looking for a second opinion. I hope I have helped.
UNTIL AND UNLESS A RETENTION AGREEMENT IS EXECUTED AND DELIVERED BY BOTH PARTIES TO THIS EMAIL, NO ATTORNEY-CLIENT RELATIONSHIP IS ESTABLISHED BETWEEN ME AND THE PERSON OR PERSONS POSING THESE QUESTIONS.