[R-package] Ensure print.lgb.Booster() works when objective is not explicitly set (#6850)

* Handling NULL objective when booster is printed, moving test functions out of method tests, creating unique tests for different method situations

* Correcting whitespace

* Adding types for integer slices

* Moving test functions nearer to relevant tests, setting print to be default

* Update R-package/tests/testthat/test_lgb.Booster.R

---------
Co-authored-by: James Lamb <jaylamb20@gmail.com>
Co-authored-by: Nikita Titov <nekit94-08@mail.ru>

R-package/R/lgb.Booster.R

@@ -1235,6 +1235,9 @@ print.lgb.Booster <- function(x, ...) {
 
   if (!handle_is_null) {
     obj <- x$params$objective
+    if (is.null(obj)) {
+      obj <- "(default)"
+    }
     if (obj == "none") {
       obj <- "custom"
     }

R-package/tests/testthat/test_lgb.Booster.R

@@ -1518,25 +1518,24 @@ test_that("boosters with linear models at leaves can be written to RDS and re-lo
     expect_identical(preds, preds2)
 })
 
-test_that("Booster's print, show, and summary work correctly", {
-    .have_same_handle <- function(model, other_model) {
+.have_same_handle <- function(model, other_model) {
   expect_equal(
     model$.__enclos_env__$private$handle
     , other_model$.__enclos_env__$private$handle
   )
-    }
+}
 
-    .has_expected_content_for_fitted_model <- function(printed_txt) {
+.has_expected_content_for_fitted_model <- function(printed_txt) {
   expect_true(any(startsWith(printed_txt, "LightGBM Model")))
   expect_true(any(startsWith(printed_txt, "Fitted to dataset")))
-    }
+}
 
-    .has_expected_content_for_finalized_model <- function(printed_txt) {
+.has_expected_content_for_finalized_model <- function(printed_txt) {
   expect_true(any(printed_txt == "LightGBM Model"))
   expect_true(any(grepl("Booster handle is invalid", printed_txt, fixed = TRUE)))
-    }
+}
 
-    .check_methods_work <- function(model) {
+.check_methods_work <- function(model) {
 
    #--- should work for fitted models --- #
 
@@ -1584,8 +1583,9 @@ test_that("Booster's print, show, and summary work correctly", {
    })
    .have_same_handle(ret, model)
    .has_expected_content_for_finalized_model(log_txt)
-    }
+}
 
+test_that("Booster's print, show, and summary work correctly for built-in objectives", {
     data("mtcars")
     model <- lgb.train(
         params = list(
@@ -1616,9 +1616,9 @@ test_that("Booster's print, show, and summary work correctly", {
         , nrounds = 5L
     )
     .check_methods_work(model)
+})
 
-
-    # with custom objective
+test_that("Booster's print, show, and summary work correctly for custom objective", {
     .logregobj <- function(preds, dtrain) {
         labels <- get_field(dtrain, "label")
         preds <- 1.0 / (1.0 + exp(-preds))
@@ -1638,6 +1638,7 @@ test_that("Booster's print, show, and summary work correctly", {
         ))
     }
 
+    data("iris")
     model <- lgb.train(
         data = lgb.Dataset(
             as.matrix(iris[, -5L])
@@ -1653,6 +1654,24 @@ test_that("Booster's print, show, and summary work correctly", {
     .check_methods_work(model)
 })
 
+test_that("Booster's print, show, and summary work correctly when objective is not provided", {
+  data("iris")
+  model <- lgb.train(
+      data = lgb.Dataset(
+          as.matrix(iris[, seq_len(3L)])
+          , label = iris[, 4L]
+      )
+      , verbose = .LGB_VERBOSITY
+      , nrounds = 5L
+      , params = list(num_threads = .LGB_MAX_THREADS)
+  )
+
+  log_txt <- capture.output(print(model))
+  expect_true(any(log_txt == "Objective: (default)"))
+
+  .check_methods_work(model)
+})
+
 test_that("LGBM_BoosterGetNumFeature_R returns correct outputs", {
     data("mtcars")
     model <- lgb.train(