I'm doing a similar work related to patients' "death" in the remote patient monitoring setting. My thoughts based on your context are the following:
In your case, the study period or say observation time is 14 days since we only observe each customer's behavior during their free trial time, and then we want to understand whether a customer will "survival" (buy this product, start subscribing to the membership, or any other actions that represent this customer is willing to continue/extend/stay etc). The definition of event here could be "leaving early" as Isabella Ghement said, not using the product very frequently, e.g., if a customer still don't subscribe the product even after the 14 free trail days, then event=0 and survival_time=14+ where "+" means maybe the customer will subscribe in the future but we don't know the specific date of subscribing (right-censoring).
Since we just observe everyone's first 14 days of behavior, variables could include like # of log in time, # of click times, # of comment leaving times, # of response if comments are left, # of app opening times, the time of all these actions during the day(morning/afternoon/evening/midnight), etc... lots of variables you can include.
As for algorithms and metrics, Kaplan-Meier, CoxPH regression, Random Survival Forest, Conditional Survival Forest , XGBoost accelerated failure time, Deep Surv, Deep Hit, etc... C-index and cumulative dynamic AUC can be used for model performance, and libraries include like scikit-survival, pysurvival, lifelines. But first, the data must be of good quality.
Some extra thinkings, this type of problem can also switch to a binary classification problem. But, if you want to know like when a customer will start subscribing or stop using during this 14 free trial days, then go with survival analysis, though I think survival analysis is still not so mature compared to traditional regression/classification problem.
Hope this helps and happy to discuss more! :)