How to interpret an ACF and PACF together? I am using for the first time the tools of autocorrelation and partial autocorrelation and I am facing some difficulties regarding the interpretation of the results. (Maybe I misunderstood what partial autocorrelation was?)
To illustrate my problem, I use the number of daily views on the famous Wikipedia site between 01/01/2017 and 31/12/2021 (data available here).
Here is a graph of the raw data:

We can clearly observe an annual cycle with a significant drop in views during the summer. The time series is too large, but we can also assume that there is a weekly cycle with an increase in the number of views on weekends. To highlight these different cycles, I have used an autocorrelation.
Here are the results :

We observe a very strong correlation if the lag is 1 day, the number of views on day d is thus "influenced" by the number of views on day d-1. But we also see very strong correlations for lags of 7 days and multiples of 7 days, implying the existence of a weekly cycle. Same for a 365 days lag (actually, the correlation for a 364 days lag is more important, but I guess it is because of the influence of the weekly cycle, 364 being divisible by 7), implying the existence of an annual cycle.
However, the correlations appear to be statistically significant for a (too) large number of values. To see if the correlation values for multiple lags of 7 days (14 d, 21 d, 28 d, etc) are only high because they are multiples of 7 or if on the contrary there are really cycles of 14 d, 21 d, 28 d etc, one can use (from what I understand) a partial autocorrelation.
Here are the results:

And here, I confess I don't know how to interpret the results... Certainly, we no longer see correlation peaks for multiples of 7, confirming that there are no 14 d, 21 d, 28 d cycles. However, the partial autocorrelation also seems to show that there is no clear annual cycle (few values around 365 are statistically significant). Another surprise is that the partial autocorrelation for a 6-day lag is significantly higher than that for a 7-day lag, whereas the opposite was true for the total autocorrelation.
Have I misunderstood how partial and total autocorrelation works? Or is it simply not a suitable tool in this case? If not, can I still conclude that there is a weekly cycle and an annual cycle, even though for the latter the partial autocorrelation seems to say otherwise?

Let us clear a few definitions first, before we dive into your answer.
You often say cycle instead fo season. A cycle is something different than a season. A season is an effect clearly recurring within a year. Everything that goes beyond a year and may suddenly change its behavior strongly, is a cycle. You can describe it as a self strengthening season that suddenly drops.
For example, you have a rising population of some animals in summer, due to the fact that there are more, they reproduce next year also but on a much higher level. They do this until they do not find enough to eat or get eaten, then their cycle drops and the rising of the population starts a new. So there is no fixed frequency.
See a good example in Rob Hyndmans Book: Forecasting: Principles and Practice, Example in Book
Regarding ACF and PACF.
Look at my picture I have drawn for you, and it borrows from that video: ritvikmath
The ACF is the Correlation of two time series, where one time series, is, for example, shifted by two days. To get this effect, you take a direct (green) and indirect route (pink). Both effects together form the ACF or the correlation of these both series.
The PACF only looks at the direct route effect. For this we 'get rid' of the indirect route, because the high correlation or good values of an ACF may only be because of that indirect route. We want a clear sight on the PACF to be sure, there is a time lagged effect, and not something that piled up over time.

Now looking at YOUR PACF plot compared to ACF you no longer find the effects, you found in the ACF. That clearly means, that you thought you had a high correlation, but that was just a piled up effect. Just looking at the PACF, we can see now which factors would be good in a model because they would help us for predicting the views. And in your case all coefficients that are out of the bounds.
And with this you would get a good AR model.
Maybe you now wonder, why draw an ACF Plot at all, if PACF is so much better? Well PACF is good for identfying the autoregressive part (AR). If you want to forecast views with views from the past then an AR model is good.
The ACF is good when looking at the MA part the Moving Average of your Model. Or so to speak you do not look at the values of past views, but on the errors of your past forecast for views. And you correct your forecast by the error.
That your ACF goes up again, may be a sign of, that your time series is not stationary:
See also:
ACF Plot normally degrades to 0
