Binomial GLM in R: what do the coefficients talk about?

Question

I'd like to uderstand what the coefficents of the binomial GLM models are. In my hypothetical data:

# Create the dataset
set.seed(1)
n <- 50
cov <- 10
x <- c(rep(0,n/2), rep(1, n/2))
p <- 0.4 + 0.2*x
y <- rbinom(n, cov, p)

Now we fit a logistic regression model with x as a covariate, using the logit link:

model0 <- glm(cbind(y, cov-y) ~ x, family="binomial")
summary(model0)
# Call:
# glm(formula = cbind(y, cov - y) ~ x, family = "binomial")

# Deviance Residuals: 
#      Min        1Q    Median        3Q       Max  
# -1.50013  -0.58688  -0.05123   0.48348   2.43452  

# Coefficients:
#             Estimate Std. Error z value Pr(>|z|)    
# (Intercept)  -0.3064     0.1280  -2.394 0.016668 *  
# x             0.6786     0.1815   3.739 0.000185 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# (Dispersion parameter for binomial family taken to be 1)

#     Null deviance: 53.720  on 49  degrees of freedom
# Residual deviance: 39.537  on 48  degrees of freedom
# AIC: 177.99

# Number of Fisher Scoring iterations: 4

Returning to the original data scale

model_intercept <- 1/(1+(1/(exp(model0[[1]][1]))))
model_intercept
# (Intercept) 
#       0.424
model_x <- 1/(1+(1/(exp(model0[[1]][2]))))
model_x
#         x 
# 0.6634292 

And here from me starting a big confusion. The 0.6634292 values mean that the y increases at a rate of 0.6634292 per unit of x. Or 0.6634292 means 66,34292% mean increase rate of y with x interval used. And about the intercept? Something like despite x the y starts at 0.4 units, despite the negative value in the binomial model.

Thanks in advance for your time and help.