Just to be clear to start, beta is a statistical property. So if your beta is 0.8 over a period of time. Stock X moved on average 0.8 for a point move in the index. We might hope this property is persistent and it seems to be fairly persistent (predictable) but it doesn't have to be.
Also it is important to note this is not a lag in time. Beta is a measure of the average size of a move in the stock at the same time as a move in the index. In your example both the stock and index are measured at end of day. You can say that the stock "lags" behind the index because it doesn't grow as quickly as the market when the market is growing, but this is not a lag in time just a lag in magnitude.
People do occasionally calculate betas between a stock and lagged in time market prices, but this is not the commonly used meaning of beta. This might actually be a more useful measure as then you could bet on the future of the stock given what happened today in the market, but these "betas" tend to be much more unstable than the synchronized version and hard to trade on.
When you calculated beta you choose a time scale, in this case daily. So if your calculation is on a day-to-day basis then you have only tested the relationship on a day-to-day basis not, for instance, on a week-to-week basis. Now day-to-day and week-to-week betas are often related and are generally reasonably close but they do not have to be. There can be longer term effects only picked up on the longer scale. Stock X could day-to-day with a (average) beta of 1 to the stock market, but could have even a negative beta year-to-year with the market if the stock is counter-cyclical to longer scale trends on the market. So beta can change with the time scale used in the calculation.