Right-censored independent variable in Cox/logistic regression

Question

I have a right-censored continuous independent variable that I want to include in a Cox regression. The variable is a physiologic test which is capped at a certain time, say 120 seconds, due to safety reasons, while the time endured is the measure. 70 % of the sample endure the whole test and the variable is therefore right-censored.

How should this variable be treated in the regression? The variable, as it is, has a linear association with my outcome.

Based on several posts (e.g. here or here) I conclude that I need to include an indicator variable of whether a person is censored or not.

So this is what I have done:

1. Created an indicator variable, call it CENSORED, which is 1 for those who reached the maximum of 120 s, and 0 if else.
2. I have recoded my continuous independent variable, call it TEST, as 0 if CENSORED = 1. The value of 0 is arbitrarily chosen.
3. I have included CENSORED and TEST simultaneously in the regression.

Am I good so far?

Most of the previous posts I have found deal with linear regression. My question is how to interpret the hazard ratios (HR) (or odds ratios (OR) in logistic regression)?

My interpretation is as follows:

1. The HR for TEST is the difference in risk of my outcome for a one-unit difference in TEST (= 1 second) among those who where not censored, i.e. CENSORED = 0.
2. The HR for CENSORED is the difference in risk for being censored, i.e. CENSORED = 1, compared to TEST = 0, assuming a linear association. The reference level of TEST = 0 is chosen in the recoding of the variable, and should reflect a meaningful contrast to being censored. For now I chose TEST = 0, i.e. the time of the test is 0.

Is this correct, or is the HR for the indicator variable without relevant meaning?

I also checked the linearity assumption of the original continuous IV among CENSORED = 0; the assumption of linearity seems to be less clear here than for the whole variable, but the assumption still seems to hold.

Edit: I have not yet figured this out but another solution will simply be to use the CENSORED variable alone in the Cox regression, with the immediate interpretation. This also makes it easier to check the proportional hazard assumption, as I am not sure how it should be checked when both the continuous variable and the indicator variable is included.

(If someone would like, please provide the formula for the calculation of the HR, or perhaps easier the OR (as I also have the possibility to do logistic regression) based on the regression equation:

logit P(D=1|x) = α + β₁TEST + β₂CENSORED)