Have I understood logictic regression output correctly? Logit, odds ratio and probability ratio Here is an example model using diamonds dataset:
library(tidyverse)
library(caret)
library(broom)

# data
my_diamonds <- diamonds
my_diamonds <- my_diamonds %>% 
  mutate(target = ifelse(cut == "Fair" | cut == "Good", 0, 1))


# statistical modeling with logistic regression via caret
## tuning & parameters
set.seed(123)
train_control <- trainControl(
  method = "cv",
  number = 5,
  savePredictions = TRUE,
  verboseIter = TRUE,
  classProbs = TRUE,
  summaryFunction = prSummary
)

# fit model
linear_model = train(
  x = select(my_diamonds, c(price, x, y, z)), 
  y = my_diamonds$target %>% make.names() %>% factor(levels = c("X1", "X0")),
  trControl = train_control,
  method = "glm", # logistic regression
  family = "binomial",
  metric = "AUC" # actually prAUC
)

# get coefficients and various ratios
model_df <- broom::tidy(linear_model$finalModel) %>% 
  select(term, estimate) %>% 
  mutate(odds_ratio = exp(estimate))

Here is the output of model_df:
model_df
# A tibble: 5 x 3
  term         estimate odds_ratio
  <chr>           <dbl>      <dbl>
1 (Intercept) -5.81        0.00299
2 price       -0.000201    1.000  
3 x           -1.72        0.179  
4 y           -2.45        0.0867 
5 z            7.99     2959.     

Within the context of inference and understanding, can I infer the following:


*

*For inference, the estimate is not so interesting, it's the logit which is scaled log odds between 0 and 1. Typically, of more interest is the log odds... True or False?

*The log odds, calculated above with exp(estimate) are the change in odds relative to the intercept of each coefficient. Example, in this model price has a log odds ratio of exactly 1, so the change in odds per one unit change in price is unchanged, since it changes by a factor of 1 only. i.e. if price is 10 then the odds of target being 1 are still just 0.00299 all else being equal. If the log odds for price were 2 and not 1, then the odds of target being 1 when price is 10 would be (0.00299*2)^10. Is this correct?

*Each time input variable x increases by one, the log odds of target being 1 increases by 0.179 * -1.72 = −0.30788.

*For understanding actual probabilities, I was initially tempted to do this:
# don't do this because the probability ratio is not linear, must be applied to a given observation not the coefficients
model_df_dont_do_this <- model_df %>% mutate(probs_ratio = exp(estimate)/(1 + exp(estimate)))
Looks like this:
model_df_dont_do_this
# A tibble: 5 x 4
  term         estimate odds_ratio probs_ratio
  <chr>           <dbl>      <dbl>       <dbl>
1 (Intercept) -5.81        0.00299     0.00298
2 price       -0.000201    1.000       0.500  
3 x           -1.72        0.179       0.152  
4 y           -2.45        0.0867      0.0798 
5 z            7.99     2959.          1.000  

However, from researching on this site and elsewhere, this is nonsensical since the probability ratio is not linear and can only be applied to a given observation, not to the coefficients. Is that correct?
I'm seeking validation on my understanding. Have I understood logistic regression correctly? If not, where have I misunderstood?
 A: You are correct in your statement "this is nonsensical since the probability ratio is not linear and can only be applied to a given observation, not to the coefficients".  Even what you call "odds ratio" are not quite coefficients, though they are useful 
I cannot actually get your linear_model to work, though 
fit1 <- glm(target ~ price + x + y + z, family=binomial, data=my_diamonds)
fit1$coefficients
exp(fit1$coefficients)
exp(fit1$coefficients) / (1 + exp(fit1$coefficients))

produces the same coefficients as you have up to rounding, though with the "log-odds coefficients" signs reversed, the "odds coefficients" as the reciprocals of yours, and the "probability coefficients" $1$ minus your values  (probably something to do with the translation to factors and their use, so your log-odds, odds and probabilities may be that the diamonds are Fair or Good).   That point does not affect your actual question
For the first diamond, with a price of $326$, $x=3.95$, $y=3.98$, $z=2.43$, you can say that the model's log-odds are about $$-5.81 -0.000201 \times 326 -1.72 \times 3.95 -2.45 \times 3.98 + 7.99 \times 2.43 \approx -3$$
and the model's odds for this diamond are about $$0.00299 \times 1.000^{326} \times 0.179^{3.95} \times 0.0867^{3.98}  \times 2959^{2.43} \approx 0.05$$
which is not much of a surprise as $e^{-3}\approx 0.05$
But there is no equivalent simple interpretation on how to use your column labelled probs_ratio to combine with the known values for this diamond to give a model probability also about $0.05$ 
